Abstract:
A method of authentication of a terminal generating a magnetic field by a transponder including an oscillating circuit from which a D.C. voltage is generated, wherein at least one quantity depending on the coupling between the transponder and the terminal is compared with at least one reference value.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the priority benefit of French patent application Ser. No. 09/54345, filed on Jun. 25, 2009, entitled “AUTHENTICATION OF A TERMINAL BY AN ELECTROMAGNETIC TRANSPONDER,” which is hereby incorporated by reference to the maximum extent allowable by law. 
       BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention generally relates to electronic systems, and more specifically to systems using electromagnetic transponders, that is, transceivers capable of being interrogated in a contactless and wireless manner by a read and/or write terminal. 
         [0004]    2. Discussion of the Related Art 
         [0005]    Many communication systems are based on a modulation of an electromagnetic field generated by a terminal. They range from the simplest electronic tag used as a theft-prevention device to more complex systems where a transponder intended to communicate with the terminal when it is in its field, is equipped with calculation functions (electronic purse, for example) or data processing functions. 
         [0006]    Electromagnetic transponder systems are based on the use of oscillating circuits comprising a winding forming an antenna, on the transponder side and on the terminal side. Such circuits are intended to be coupled by a near magnetic field when the transponder enters the field of the terminal. The oscillating circuits of the terminal and of the transponder are generally tuned to the same frequency corresponding to the excitation frequency of the oscillating circuit of the terminal. 
         [0007]    In most cases, transponders have no autonomous power supply and extract the power supply necessary to their circuits from the high-frequency field radiated by the antenna of the terminal. 
         [0008]    When a transponder needs to communicate with a terminal, the transponder may have to authenticate the terminal before accepting a data exchange. For example, in applications where the transponder is used as a payment means (be it in money or units of account), it may reserve payments to certain terminals. According to another example, a transponder of chip card type, associated with electronic equipment (for example, a personal digital assistant or a cell phone) identifies or authenticates a user in a communication with other electronic equipment (for example, a laptop or desktop computer). 
         [0009]    Currently, authentication processes use cryptography algorithms and a data exchange between the terminal and the transponder. Such processes require significant power- and time-intensive calculations. Further, any cryptographic process is more or less sensitive to attacks aiming at discovering the secret of the authentication to hack the system. 
       SUMMARY OF THE INVENTION 
       [0010]    It would be desirable for a transponder to be able to authenticate a terminal with which it needs to communicate before it has to transmit data to the terminal. 
         [0011]    It would also be desirable to have an authentication process independent from any cryptography. 
         [0012]    It would also be desirable to have a fast, less power- and calculation-intensive authentication process. 
         [0013]    It would also be desirable to authenticate the terminal without having to perform, on each authentication, a data exchange between the terminal and the transponder. 
         [0014]    To achieve all or part of these objects as well as others, at least one embodiment of the present invention provides a method of authentication of a terminal generating a magnetic field, by a transponder comprising an oscillating circuit from which a D.C. voltage is generated, wherein at least one quantity depending on the coupling between the transponder and the terminal is compared with at least one reference value. 
         [0015]    According to an embodiment of the present invention, a quantity corresponding to the level of said D.C. voltage is measured and compared with a reference value depending on this same voltage in a condition of optimum coupling between the transponder and the terminal. 
         [0016]    According to an embodiment of the present invention, said value in the optimum coupling condition is deduced from a training phase. 
         [0017]    According to an embodiment of the present invention: 
         [0018]    a first quantity corresponding to the level of said D.C. voltage is measured and stored for a first value of the resistive load formed by circuits of the transponder on the oscillating circuit; 
         [0019]    a second quantity corresponding to the level of said D.C. voltage is measured and stored for a second value of the resistive load on the oscillating circuit; and 
         [0020]    a ratio between the two quantities is compared with one or several reference values. 
         [0021]    According to an embodiment of the present invention, said reference value depends on the respective resistive load values. 
         [0022]    According to an embodiment of the present invention, a variation of the resistive load between the first and second values is obtained by modifying the consumption of processing circuits comprised by the transponder. 
         [0023]    According to an embodiment of the present invention, a variation of the resistive load between the first and second values is obtained by switching a resistive retromodulation element comprised by the transponder. 
         [0024]    According to an embodiment of the present invention, in the absence of an authentication, the transponder sends intentionally incorrect data. 
         [0025]    The present invention also provides an electromagnetic transponder comprising: 
         [0026]    an oscillating circuit upstream of a rectifying circuit capable of providing a D.C. voltage when the transponder is present in the magnetic field of a terminal; and 
         [0027]    at least one processing unit capable of implementing the authentication method. 
         [0028]    The foregoing objects, features, and advantages of the present invention will be discussed in detail in the following non-limiting description of specific embodiments in connection with the accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0029]      FIG. 1  is a very simplified representation of a transponder system of the type to which the present invention applies as an example; 
           [0030]      FIG. 2  is a simplified block diagram of a terminal and of a transponder of an electromagnetic transponder communication system; 
           [0031]      FIG. 3  illustrates an example of the variation of the voltage across the oscillating circuit of the transponder according to the coupling factor; 
           [0032]      FIG. 4  is a functional block diagram illustrating an embodiment of the method of terminal authentication by a transponder; 
           [0033]      FIG. 5  is a functional block diagram illustrating an embodiment of a training phase; and 
           [0034]      FIG. 6  is a block diagram of an embodiment of a transponder capable of authenticating a terminal. 
       
    
    
     DETAILED DESCRIPTION 
       [0035]    The same elements have been designated with the same reference numerals in the different drawings. For clarity, only those steps and elements which are useful to the understanding of the present invention have been shown and will be described. In particular, the communications between the transponder and the terminal have not been detailed, the present invention being compatible with any usual communication. Further, the functions capable of being implemented by a terminal or by a transponder, other than the authentication by this transponder, have not been detailed either, the present invention being here again compatible with any usual function of a terminal or of a transponder. 
         [0036]      FIG. 1  is a block diagram of an electromagnetic transponder communication system. A terminal  1  (TERMINAL) is capable of communicating in near field (for example according to a near field communication protocol NFC) with a distant element, that is, a transponder (TRANS). 
         [0037]    The terminal may take different forms, for example, a transport ticket validation terminal, an electronic passport reader, a laptop computer, a mobile telecommunication device (GSM phone, PDA, etc.), an electronic control unit for starting an automobile vehicle, etc. 
         [0038]    The transponder may similarly take different forms, for example, a chip card, an electronic transport ticket, an electronic passport, a telecommunication terminal (GSM phone, PDA, etc.), an electronic tag, etc. 
         [0039]      FIG. 2  very schematically shows a simplified example of a terminal  1  and of a transponder  2 . 
         [0040]    Terminal  1  comprises an oscillating circuit, generally series, formed of an inductance L 1  in series with a capacitor C 1  and a resistor R 1 . This series oscillating circuit is, in the example of  FIG. 2 , connected between an output terminal  12  of an amplifier or antenna coupler  14  and a terminal  13  at a reference voltage (generally the ground). An element  15  for measuring the current in the oscillating circuit is interposed, for example, between capacitive element C 1  and ground  13 . Measurement element  15  belongs to a phase regulation loop which will be described hereafter. Amplifier  14  receives a high-frequency transmission signal originating from a modulator  16  (MOD) which receives a reference frequency (signal OSC), for example, from a quartz oscillator (not shown). Modulator  16  receives, if need be, a signal Tx originating from a circuit  11  for controlling and exploiting the transmissions. Circuit  11  is generally provided with a control and data processing microprocessor, communicating with different input/output circuits (keyboard, display, element of exchange with a server, etc.) and/or processing circuits, not shown. The elements of terminal  1  most often draw the power necessary to their operation from a supply circuit (not shown) connected, for example, to the power line distribution system (mains) or to a battery (for example, that of an automobile vehicle or of a portable telephone or computer). Modulator  16  provides a high-frequency carrier (for example, at 13.56 MHz) to series oscillating circuit L 1 -C 1 , which generates a magnetic field. 
         [0041]    Capacitive element C 1  is, for example, a variable-capacitance element controllable by a signal CTRL. This element takes part in the phase regulation of current I 1  in antenna L 1  with respect to a reference signal. This regulation is a regulation of the high-frequency signal, that is, of the signal of the carrier corresponding to the signal provided to amplifier  14  in the absence of data Tx to be transmitted. The regulation is performed by varying capacitance C 1  of the oscillating circuit of the terminal to maintain the current in the antenna in constant phase relationship with a reference signal. This reference signal for example corresponds to signal OSC provided to modulator  14 . Signal CTRL originates from a circuit  17  (COMP) having the function of detecting the phase interval with respect to the reference signal and of accordingly modifying the capacitance of element C 1 . The comparator receives data MES about current I 1  in the oscillating circuit detected by measurement element  15  (for example, a current transformer or a resistor). 
         [0042]    A transponder  2 , capable of cooperating with terminal  1 , comprises an oscillating circuit, for example, parallel, formed of an inductance L 2  in parallel with a capacitor C 2  between two terminals  21  and  22 . The parallel oscillating circuit (called receive mode resonant circuit) is intended to capture the magnetic field generated by oscillating circuit L 1 -C 1  of terminal  1 . Circuits L 2 -C 2  and L 1 -C 1  are tuned to a same resonance frequency (for example, 13.56 MHz). Terminals  21  and  22  are connected to two A.C. input terminals of a rectifying bridge  23  (most often, fullwave). The rectified output terminals of bridge  23  respectively define a positive terminal  24  and a reference terminal  25 . A capacitor Ca is connected between terminals  24  and  25  to smooth the rectified voltage. The recovered power is used to recharge a battery, not shown. 
         [0043]    When transponder  2  is in the field of terminal  1 , a high-frequency voltage is generated across resonant circuit L 2 -C 2 . This voltage, rectified by bridge  23  and smoothed by capacitor Ca, provides a supply voltage to electronic circuits of the transponder via a voltage regulator  26  (REG). Such circuits generally comprise a processing unit  27  (for example, a microcontroller μC) associated with a memory (not shown), a demodulator  28  (DEM) of the signals that may have been received from terminal  1 , and a modulator  29  (MOD) for transmitting data to the terminal. The transponder is generally synchronized by means of a clock (CLK) extracted, by a block  20 , from the high-frequency signal recovered, before rectification, from one of terminals  21  and  22 . Most often, all the electronic circuits of transponder  2  are integrated in a same chip. 
         [0044]    To transmit data from terminal  1  to the transponder, circuit  16  modulates (generally in amplitude) the carrier (signal OSC) according to signal Tx. On the side of transponder  2 , these data are demodulated by demodulator  28  based on voltage V Ca . The demodulator may sample the signal to be demodulated upstream of the rectifying bridge. 
         [0045]    To transmit data from transponder  2  to terminal  1 , modulator  29  controls a stage  30  of modulation (retromodulation) of the load formed by the transponder circuits on the magnetic field generated by the terminal. This stage is generally formed of an electronic switch K 30  (for example, a transistor) and of a resistor R 30  (or a capacitor), in series between terminals  24  and  25 . Switch K 30  is controlled at a so-called sub-carrier frequency (for example, 847.5 kHz), much lower (generally with a ratio of at least 10) than the frequency of the excitation signal of the oscillating circuit of terminal  1 . When switch K 30  is on, the oscillating circuit of the transponder is submitted to an additional damping with respect to the load formed by circuits  20 ,  26 ,  27 ,  28 , and  29  so that the transponder samples a greater amount of power from the high-frequency magnetic field. On the side of terminal  1 , amplifier  14  maintains the amplitude of the high-frequency excitation signal constant. Accordingly, the power variation of the transponder translates as an amplitude and phase variation of the current in antenna L 1 . This variation is detected by an amplitude or phase demodulator of the terminal. In the embodiment illustrated in  FIG. 2 , comparator  17  integrates a phase demodulator also used to demodulate the signal originating from the transponder. Accordingly, comparator  17  provides a signal Rx giving back to circuit  11  a possible retromodulation of data received from a transponder. Other demodulation circuits may be provided, for example, a circuit exploiting a measurement of the voltage across capacitor C 1 . 
         [0046]    Many variations exist to encode/decode and modulate/demodulate communications between a transponder and a terminal. 
         [0047]    The response time of the phase regulation loop is sufficiently long to avoid disturbing the possible retromodulation from a transponder and sufficiently short as compared with the speed at which a transponder passes in the field of the terminal. One can speak of a static regulation with respect to the modulation frequencies (for example, the 13.56-MHz frequency of the remote supply carrier and the 847.5-kHz retromodulation frequency used to transmit data from the transponder to the terminal). 
         [0048]    An example of a phase regulation terminal is described in document EP-A-0857981. 
         [0049]    Regulating the phase on the terminal side enables allows current and voltage measurements in the oscillating circuit of the transponder to deduce from these measurements information relative to the transponder coupling when it is in the field of the terminal. The coupling coefficient between the oscillating circuit of the terminal and of the transponder essentially depends on the distance separating the transponder from the terminal. The coupling coefficient, noted k, is always between 0 and 1. It can be defined by the following formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     = 
                     
                       M 
                       
                         
                           L 
                            
                           
                               
                           
                            
                           
                             1 
                             · 
                             L 
                           
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
         [0050]    where M represents the mutual inductance between inductances L 1  and L 2  of the oscillating circuits of the terminal and of the transponder. 
         [0051]    An optimum coupling is defined as being the position at which voltage V C2  across the oscillating circuit of the transponder is maximum. This optimum coupling, noted k opt , may be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       k 
                       opt 
                     
                     = 
                     
                       
                         
                           
                             L 
                              
                             
                                 
                             
                              
                             2 
                           
                           
                             L 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         · 
                         
                           
                             R 
                              
                             
                                 
                             
                              
                             1 
                           
                           
                             R 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0052]    where R 2  represents the resistance equivalent to the load formed by the elements of the transponder on its own oscillating circuit. In other words, resistance R 2  represents the equivalent resistance of all the circuits of transponder  2 , placed in parallel on capacitor C 2  and inductance L 2  (before or after the rectifying bridge). The conductance due to the transponder circuits will be called “resistive load”. The level of this load is symbolized by resistor R 2  in parallel across the oscillating circuit. In above formula 2, the series resistance of inductance L 1  (terminal antenna) has been neglected. It can also be considered that the value of this series resistance is, for simplification, included in the value of resistor R 1 . 
         [0053]    Formula 2 represents a signature of the terminal-transponder couple. For a same transponder and given operating conditions (load R 2 ), the optimum coupling coefficient varies according to the terminal which conditions values L 1  and R 1 . 
         [0054]    It is provided to take advantage of this feature to enable a transponder to authenticate the terminal in the range of which it is located by indirectly verifying this signature. 
         [0055]      FIG. 3  shows an example of the variation of voltage V C2  recovered on the transponder side according to the coupling k/k opt  normalized with respect to the optimum coupling. The curve starts from the origin of ordinates (zero voltage) for a zero coupling. This corresponds to a distance from the transponder to the terminal such that no signal is sensed by the transponder. Voltage V C2  reaches a maximum value V C2opt  for optimum coupling coefficient k opt  (k/k opt =1), then decreases to an intermediary value V C2 (1) reached at coupling k=1. 
         [0056]    To evaluate, on the transponder side, the coupling of this transponder with the terminal, the information of voltage V C2  across capacitive element C 2  of its oscillating circuit is exploited. This voltage is provided by the following relation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                       
                     
                     = 
                     
                       
                         I 
                          
                         
                             
                         
                          
                         2 
                       
                       
                         ω 
                         · 
                         
                           C 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
         [0057]    where I 2  represents the current in the oscillating circuit of the transponder, and where ω represents the pulse of the signal. 
         [0058]    Current I 2  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                        
                       
                           
                       
                        
                       2 
                     
                     = 
                     
                       
                         
                           M 
                           · 
                           ω 
                           · 
                           I 
                         
                          
                         
                             
                         
                          
                         1 
                       
                       
                         Z 
                          
                         
                             
                         
                          
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
         [0059]    where I 1  represents the current in the oscillating circuit of the terminal and where Z 2  represents the transponder impedance. 
         [0060]    Impedance Z 2  of the transponder is provided by the following relation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                        
                       
                           
                       
                        
                       
                         2 
                         2 
                       
                     
                     = 
                     
                       
                         X 
                          
                         
                             
                         
                          
                         
                           2 
                           2 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               L 
                                
                               
                                   
                               
                                
                               2 
                             
                             
                               R 
                                
                               
                                   
                               
                                
                               2. 
                                
                               C 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
         [0061]    where X 2  represents the imaginary part of the impedance of the oscillating circuit 
         [0000]    
       
         
           
             
               ( 
               
                 
                   X 
                    
                   
                       
                   
                    
                   2 
                 
                 = 
                 
                   
                     
                       ω 
                       · 
                       L 
                     
                      
                     
                         
                     
                      
                     2 
                   
                   - 
                   
                     1 
                     
                       
                         ω 
                         · 
                         C 
                       
                        
                       
                           
                       
                        
                       2 
                     
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0062]    Further, current I 1  in the oscillating circuit of the terminal is given by the following relation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                        
                       
                           
                       
                        
                       1 
                     
                     = 
                     
                       Vg 
                       
                         Z 
                          
                         
                             
                         
                          
                         
                           1 
                           app 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
         [0063]    where Vg designates a so-called generator voltage, exciting the oscillating circuit of the terminal, and where Z 1   app  represents the apparent impedance of the oscillating circuit. 
         [0064]    The fact of regulating the phase of the oscillating circuit of the terminal to enables for all the variations which would tend to modify, statically with respect to the modulation frequencies, the imaginary part of the load formed by the transponder, to be compensated by the phase regulation loop. It is thus ensured that in static operation, the imaginary part of impedance Z 1   app  is zero. Accordingly, impedance Z 1   app  becomes equal to apparent resistance R 1   app  (real part of the impedance) and may be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                      
                     
                         
                     
                      
                     
                       1 
                       app 
                     
                   
                   = 
                   
                     
                       R 
                        
                       
                           
                       
                        
                       
                         1 
                         app 
                       
                     
                     = 
                     
                       
                         R 
                          
                         
                             
                         
                          
                         1 
                       
                       + 
                       
                         
                           
                             
                               
                                 k 
                                 2 
                               
                               · 
                               
                                 ω 
                                 2 
                               
                               · 
                               L 
                             
                              
                             
                                 
                             
                              
                             
                               1 
                               · 
                               L 
                             
                              
                             
                                 
                             
                              
                             
                               2 
                               2 
                             
                           
                           
                             Z 
                              
                             
                                 
                             
                              
                             
                               
                                 2 
                                 2 
                               
                               · 
                               R 
                             
                              
                             
                                 
                             
                              
                             
                               2 
                               · 
                               C 
                             
                              
                             
                                 
                             
                              
                             2 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
         [0065]    Since the oscillating circuits are tuned, it can be considered that imaginary part X 2  of impedance Z 2  is, as a first approximation, close to zero. As a result, the value of impedance Z 2  can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                      
                     
                         
                     
                      
                     2 
                   
                   = 
                   
                     
                       
                         L 
                          
                         
                             
                         
                          
                         2 
                       
                       
                         R 
                          
                         
                             
                         
                          
                         
                           2 
                           · 
                           C 
                         
                          
                         
                             
                         
                          
                         2 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
         [0066]    By inserting this simplification into formulas 4 and 7, and inserting formula 4 into formula 3, the following formula can be obtained for voltage V C2  recovered across the oscillating circuit of the transponder: 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     
                       C 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     k 
                     · 
                     
                       
                         
                           L 
                            
                           
                               
                           
                            
                           1 
                         
                         
                           L 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                     · 
                     
                       
                         
                           V 
                           g 
                         
                         
                           
                             
                               R 
                                
                               
                                   
                               
                                
                               1 
                             
                             
                               R 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                           + 
                           
                             
                               k 
                               2 
                             
                             · 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
         [0067]    Formula 9 shows that, for a given terminal (fixed values of Vg, R 1 , and L 1 ) and for a fixed impedance L 2  (and thus a fixed value of C 2 ), voltage V C2  only depends on coupling k and on the resistive load (equivalent to resistor R 2 ) formed by the transponder circuits and brought in parallel on the oscillating circuit. 
         [0068]    It should be noted that formula 9 can only be applied when the oscillating circuit of transponder L 2 -C 2  is considered to be set to the tuning frequency, that is, ω·√{square root over (L 2 ·C 2 )}=1. 
         [0069]    In optimum coupling position k opt , maximum voltage V C2opt  is thus provided by the following formula (combining formulas 2 and 9): 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     
                       C 
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       opt 
                     
                   
                   = 
                   
                     
                       
                         V 
                         g 
                       
                       2 
                     
                     · 
                     
                       
                         
                           R 
                            
                           
                               
                           
                            
                           2 
                         
                         
                           R 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
         [0070]    Formula 10 shows that, for a given terminal, it is possible to store, in a training phase where the transponder is in relation with the terminal, value V C2opt]R20  associated with the transponder-terminal couple with load R 20 . To achieve this, it is enough to know values Vg and R 1  and to measure voltage V C2 . 
         [0071]    After, in operation, an authenticity test may be performed by comparing the current value of voltage V C2]R20 , with a value R 20  of resistor R 2  to value V C2opt]R20 . The current value should be smaller than or equal to the stored value. If not, this means that the terminal is set to generate a field greater than the provided field. The terminal is then considered as inauthentic (or improperly configured). There is indeed a risk for it to be a pirate terminal. 
         [0072]    By combining formulas 9 and 10 and by expressing the coupling as normalized by the optimum coupling (k/k opt ), the following expression of voltage V C2  is obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     
                       C 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     2 
                     · 
                     
                       V 
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         opt 
                       
                     
                     · 
                     
                       
                         
                           k 
                           
                             k 
                             opt 
                           
                         
                         
                           1 
                           + 
                           
                             
                               ( 
                               
                                 k 
                                 
                                   k 
                                   opt 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
         [0073]    For a given coupling value k, considering that the impedance of the oscillating circuit of the terminal does not vary and that the circuits remain tuned, the ratio between values V C2]R21  and V C2]R20  of voltage V C2 , respectively for values R 21  and R 20  of resistor R 2 , provides, according to formula 11, the following relation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         21 
                       
                     
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         20 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               k 
                               
                                 k 
                                 
                                   
                                     opt 
                                     ] 
                                   
                                    
                                   R 
                                    
                                   
                                       
                                   
                                    
                                   20 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         1 
                       
                       
                         
                           
                             ( 
                             
                               k 
                               
                                 k 
                                 
                                   
                                     opt 
                                     ] 
                                   
                                    
                                   R 
                                    
                                   
                                       
                                   
                                    
                                   20 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             R 
                              
                             
                                 
                             
                              
                             20 
                           
                           
                             R 
                              
                             
                                 
                             
                              
                             21 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
         [0074]    Formula 12 shows that by increasing the value of resistor R 2  from a first value R 20  to a second greater value R 21  (which amounts to decreasing the load of the transponder circuits on oscillating circuit L 2 -C 2 ), voltage V C2]R21  will be greater than voltage V C2]R20 . Conversely, a decrease in the value of transistor R 2  causes a decrease in recovered voltage V C2 . 
         [0075]    Accordingly, for a given terminal (fixed values of Vg and R 1 ) and a tuned system, the voltages obtained with two resistive loads (equivalent to resistors R 20  and R 21 ) can be compared. If, when value R 20  of resistor R 2  is increased (by decreasing the resistive load) up to a value R 21 , the obtained voltage V C2]R21  is not greater than initial voltage V C2]R20 , this means that the terminal is provided to generate a current Vg/R 1  smaller than the expected current, and thus that the terminal-transponder couple does not respect the expected signature. 
         [0076]    For a resistive load variation corresponding to switching from an equivalent resistance R 2  of value R 20  to a greater value R 21 , formula 12 provides: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         k 
                         
                           k 
                           
                             
                               opt 
                               ] 
                             
                              
                             R 
                              
                             
                                 
                             
                              
                             20 
                           
                         
                       
                       ) 
                     
                     2 
                   
                   = 
                   
                     
                       1 
                       - 
                       
                         
                           
                             V 
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 ] 
                               
                                
                               R 
                                
                               
                                   
                               
                                
                               21 
                             
                           
                           
                             V 
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 ] 
                               
                                
                               R 
                                
                               
                                   
                               
                                
                               20 
                             
                           
                         
                         · 
                         
                           
                             R 
                              
                             
                                 
                             
                              
                             20 
                           
                           
                             R 
                              
                             
                                 
                             
                              
                             21 
                           
                         
                       
                     
                     
                       
                         
                           V 
                           
                             
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               ] 
                             
                              
                             R 
                              
                             
                                 
                             
                              
                             21 
                           
                         
                         
                           V 
                           
                             
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               ] 
                             
                              
                             R 
                              
                             
                                 
                             
                              
                             20 
                           
                         
                       
                       - 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
         [0077]    The combination of formulas 11 and 13 provides: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         20 
                       
                     
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             opt 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         20 
                       
                     
                   
                   = 
                   
                     
                       
                         2 
                         · 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       20 
                                     
                                   
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       21 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                      
                                     
                                         
                                     
                                      
                                     20 
                                   
                                   
                                     R 
                                      
                                     
                                         
                                     
                                      
                                     21 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       20 
                                     
                                   
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       21 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         1 
                         - 
                         
                           
                             R 
                              
                             
                                 
                             
                              
                             20 
                           
                           
                             R 
                              
                             
                                 
                             
                              
                             21 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
         [0078]    For a variation of the value of resistance R 2  from a value R 20  to a smaller value R 21 , formulas 13 and 14 respectively become: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           k 
                           
                             k 
                             
                               
                                 opt 
                                 ] 
                               
                                
                               R 
                                
                               
                                   
                               
                                
                               20 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     = 
                     
                       
                         
                           
                             
                               V 
                               
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   ] 
                                 
                                  
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 21 
                               
                             
                             
                               V 
                               
                                 
                                   
                                     C 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   ] 
                                 
                                  
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 20 
                               
                             
                           
                           · 
                           
                             
                               R 
                                
                               
                                   
                               
                                
                               20 
                             
                             
                               R 
                                
                               
                                   
                               
                                
                               21 
                             
                           
                         
                         - 
                         1 
                       
                       
                         1 
                         - 
                         
                           
                             V 
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 ] 
                               
                                
                               R 
                                
                               
                                   
                               
                                
                               21 
                             
                           
                           
                             V 
                             
                               
                                 
                                   C 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 ] 
                               
                                
                               R 
                                
                               
                                   
                               
                                
                               20 
                             
                           
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     
                       13 
                       ′ 
                     
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         20 
                       
                     
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             opt 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         20 
                       
                     
                   
                   = 
                   
                     
                       
                         2 
                         · 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     R 
                                      
                                     
                                         
                                     
                                      
                                     20 
                                   
                                   
                                     R 
                                      
                                     
                                         
                                     
                                      
                                     21 
                                   
                                 
                                 - 
                                 
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       20 
                                     
                                   
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       21 
                                     
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       20 
                                     
                                   
                                   
                                     V 
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         ] 
                                       
                                        
                                       R 
                                        
                                       
                                           
                                       
                                        
                                       21 
                                     
                                   
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             R 
                              
                             
                                 
                             
                              
                             20 
                           
                           
                             R 
                              
                             
                                 
                             
                              
                             21 
                           
                         
                         - 
                         1 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     formula 
                      
                     
                         
                     
                      
                     
                       14 
                       ′ 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0079]    Thus, the value of current value V C2]R20  with voltage can also be verified. According to the direction of the variation of resistance R 2 . If this voltage does not respect one of above relations  14  and  14 ′, this means that the terminal does not need to be considered as belonging to the provided terminal-transponder couple. 
         [0080]    In practice, rather than a direct measurement of the voltage across the oscillating circuit, a measurement the smoothed voltage across capacitor V Ca  at the output of rectifying bridge  23  is performed. Voltage V Ca  is proportional to voltage V C2 . Since voltage ratios are being evaluated, it is not necessary to know the proportionality factor between voltages V C2  and V Ca . In a specific embodiment, the measurement is performed by the microprocessor. The storage of the values of the measured voltages is performed either by analog means or, preferentially, digitally over several bits, the number of which depends on the desired accuracy of analysis. 
         [0081]      FIG. 4  illustrates the implementation of an embodiment of the method for verifying the authenticity of a terminal having the transponder in its field. To V C2]R21 , knowing that it is in practice easier to measure values V Ca]R20  and V Ca]R21  of voltage V Ca  but that this changes nothing to the comparison values. 
         [0082]    It is started (block  41 , MES V C2]R20 ) by measuring and storing the voltage across capacitor C 2  with a first value R 20  of resistor R 2 . Then, (block  42 , R 20 -&gt;R 21 ), the value of the resistive element is modified towards a greater value. 
         [0083]    Then (block  43 , V C2]R21 ), voltage V C2  is measured with resistance value R 21 , which is stored. 
         [0084]    The current value V C2]R20  obtained with value R 20  is compared (block  44 , V C2]R20 ≦V C2opt]R20 ) with value V C2opt]R20  at optimum coupling k opt . It should be reminded that this value at the optimum coupling is known by the transponder and has been stored therein during a training phase where the transponder was considered as being in the field of an authentic reader. Test  44  may be performed as soon as voltage V C2]R20  has been measured (after step  41 ). 
         [0085]    If test  44  is not satisfied (output N of block  44 ), unit  27  of the transponder for example causes an error processing (block  49 , ERROR). This processing for example corresponds to a transaction denial, to a transponder reset, to a fail-soft operation (without performing the functions which are critical as to the manipulated information), etc. It may also be provided for the transponder to send messages to mislead or confuse the terminal with intentionally incorrect information, for example, messages comprising random data. Various other processings may be envisaged, for example, any error processing usually provided in the absence of an authentication by a ciphering mechanism. 
         [0086]    If test  44  is successfully passed (output Y of block  44 ), it is provided, in the embodiment of  FIG. 4 , to perform two successive tests on the value of the ratio between voltages V C2]R20  and V C2]R21 . 
         [0087]    For this purpose, a ratio r v  between the voltages measured with the resistive loads equivalent to values R 20  and R 21  of resistor R 2  is calculated and stored 
         [0000]    
       
         
           
             
               ( 
               
                 
                   block 
                    
                   
                       
                   
                    
                   45 
                 
                 , 
                 
                   
                     r 
                     v 
                   
                   = 
                   
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         21 
                       
                     
                     
                       V 
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                           ] 
                         
                          
                         R 
                          
                         
                             
                         
                          
                         20 
                       
                     
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0088]    Then (block  46 , r v &gt;1), it is verified whether this ratio is greater than 1. If not (output N of block  46 ), it is proceeded to error processing  49 . 
         [0089]    Otherwise, it is verified (block  47 , r v &lt;R 21 /R 20 ) whether this ratio is smaller than the ratio between values R 21  and R 20 . Indeed, if not (output N of block  47 ), this means that the terminal generates a field greater than the expected field. The terminal is thus not authorized (block  49 ). 
         [0090]    Finally, a last test is then performed on the value of voltage V C2]R20 , which amounts to verifying 
         [0000]    
       
         
           
             ( 
             
               
                 block 
                  
                 
                     
                 
                  
                 48 
               
               , 
               
                 
                   V 
                   
                     
                       
                         C 
                          
                         
                             
                         
                          
                         2 
                       
                       ] 
                     
                      
                     R 
                      
                     
                         
                     
                      
                     20 
                   
                 
                 = 
                 
                   
                     
                       2 
                       · 
                       
                         
                           
                             ( 
                             
                               
                                 1 
                                 
                                   r 
                                   v 
                                 
                               
                               - 
                               
                                 
                                   R 
                                    
                                   
                                       
                                   
                                    
                                   20 
                                 
                                 
                                   R 
                                    
                                   
                                       
                                   
                                    
                                   21 
                                 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 1 
                                 
                                   r 
                                   v 
                                 
                               
                             
                             ) 
                           
                         
                       
                       · 
                       
                         V 
                         
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               opt 
                             
                             ] 
                           
                            
                           R 
                            
                           
                               
                           
                            
                           20 
                         
                       
                     
                     
                       1 
                       - 
                       
                         
                           R 
                            
                           
                               
                           
                            
                           20 
                         
                         
                           R 
                            
                           
                               
                           
                            
                           21 
                         
                       
                     
                   
                   ? 
                 
               
             
             ) 
           
         
       
     
         [0000]    whether formula 14 is respected. If such is not the case (output N of block  48 ), the error processing is applied. If, conversely, all tests are validated (output Y of block  48 ), it can be considered that the terminal having the transponder in its field is authentic (block  50 , OK) and the transaction or communication can start. 
         [0091]    The tests may be performed in a different order than that indicated hereabove. However, they are preferentially performed in an order of increasing calculation complexity, which enables to more rapidly reject a terminal which is not adapted to the transponder. 
         [0092]    Further, different intermediary values (for example, ratio r v  or ratio R 21 /R 20 ) may be stored to be reused in the successive tests or, conversely, calculated on the fly. 
         [0093]    Further, the number of performed tests depends on the application, on the reliability of the desired authentication, on the calculating power of the transponder, on the available data, etc. For example, in a simplified embodiment where the values of resistors R 20  and R 21  are not determinable but where it is only known that value R 21  is greater than value R 20 , test  46  will be sufficient. It should be noted that, in this simplified embodiment, no training is necessary. The verification may however be improved by performing several verifications with more than two values of resistor R 2 . The evaluation may also be performed by decreasing the value of resistor R 2 . It will then be ascertained that this value is sufficient to preserve a sufficient value V C2]R21  of voltage V C2  to ensure a power supply of the transponder circuits. Further, the relations of tests  43  and  47  must then be inverted (r v &lt;1 and r v &gt;R 21 /R 20 ). 
         [0094]    Tolerances or ranges of acceptable values may be introduced into the tests to take into account possible operating drifts of the terminal or, in the case of a family of authorized terminals, possible acceptable dispersions among the terminals of this family. 
         [0095]    It is thus possible, based on two voltage measurements with two resistance values of the oscillating circuit of the transponder, to perform an authentication of the terminal. 
         [0096]    The reliability of this authentication may be improved by exploiting relations determined by training. 
         [0097]      FIG. 5  is a block diagram of an embodiment of a transponder  2 , equipped to automatically determine, when it is in the field of a terminal (not shown), whether this terminal is authorized. The representation of  FIG. 5  is simplified with respect to that of  FIG. 2 . In particular, the means of demodulation, retromodulation, and for obtaining the clock frequency have not been illustrated. 
         [0098]    As previously, transponder  2  is based on a parallel oscillating circuit L 2 -C 2  having its terminals  21  and  22  connected to the input terminals of a rectifying bridge  23 . An element for measuring the current Ic intended for the processing unit may be provided at the output of regulator  26 . Further, a switchable resistive circuit  40  is provided between terminals  24  and  25  of rectifying bridge  23 . For example, two resistors R 43  and R 45  are connected in parallel, each being in series with a switch K 43 , respectively K 45 . Switches K 43  and K 45  (for example, MOS transistors) are intended to be switched to implement the method for determining the coupling position. Processing unit  27  (PU) receives information about voltage V Ca  on an input MES to implement the above-described method. In the example of  FIG. 5 , when the two resistors R 43  and R 45  are functionally connected, resistor R 2  (load of the transponder circuits) has value R 20 . The disconnection of one of the resistors (for example, resistor R 43 ) increases resistance R 2  towards value R 21 . Other connections and switchings may be provided according to the embodiment of the implemented method. For example, a single switchable resistor may be used, considering that one of the two values of resistor R 2  corresponds to the resistive load of the other transponder circuits. 
         [0099]    According to a preferred embodiment, the switchable resistor corresponds to that used for a resistive retromodulation. For example, a first measurement is performed by switching the retromodulation resistor so that it is functionally in the circuit (switch K 30  in the on state in the example of  FIG. 2 ). Voltage V C2]R20  is measured. Then, switch K 30  is turned off and voltage V C2]R21  is measured. 
         [0100]    As a variation, the increase or the decrease of equivalent resistance R 2  is caused by a variation of the power consumption of the transponder circuits, typically of processing unit  27 . For example, to decrease the value of resistor R 2  (to increase the power consumption), the execution of calculations or of processings by unit  27  is triggered. An increase of equivalent resistance R 2  may also be caused by decreasing the power consumption of unit  27  by interrupting certain calculations. As a variation, the execution speed conditioned by the clock is slowed down (block  20 ). The variation of resistance R 2  is known from the time when the power consumption of different tasks to be executed by unit  27  is known. 
         [0101]    The calculations required to authenticate a terminal are sufficiently simple for their execution time to be negligible with respect to the displacement speed of a transponder in front of a terminal (and thus the variation speed of the coupling coefficient). Such is in particular the case for transponders equipped with microcontrollers executing cryptography functions in which such calculation-intensive functions are themselves executed in a duration for which it can be considered that the coupling does not vary. In other cases, the transponder remains laid on a reception surface of the terminal and the coupling thus does not vary for an even longer period. 
         [0102]    It should be noted that the authentication is performed without requiring to establish a communication with the terminal. Accordingly, a protection of the data contained in the transponder is guaranteed since a demodulation of a request (and thus an opening of the communication functions of the transponder) can only be authorized once the terminal has been authenticated. 
         [0103]      FIG. 6  is a functional block diagram of an example of a training phase. 
         [0104]    Such a phase is implemented when a terminal is desired to be matched with a transponder or a family of transponders. For example, such a phase is implemented in a transponder calibration phase at the end of the manufacturing. According to another example, the values are determined based on samples and are recorded in non-volatile memories of the transponders in a series manufacturing. According to still another example, the matching of a transponder and of a terminal is accessible by a final user to only authorize exchanges between electromagnetic devices that it selects (for example his cell phone with his laptop computer). For values Vg and R 1  to be available for the transponder, it may be provided for these values to be communicated thereto by the actual terminal in this training phase. 
         [0105]    It is started by setting (block  61 , SET) the transponder in a relation of nominal coupling with the terminal. For example, such a relation comprises laying a chip card or another type of transponder on a terminal with which it is desired to be matched. 
         [0106]    Then (block  62 , MES V C2]R20 ), voltage V C2]R20  is measured with a given resistive load of the transponder. 
         [0107]    Finally, this voltage, value R 20  of the corresponding resistor, and values R 1  and Vg corresponding to the terminal are stored. These values may be provided by the manufacturer of the terminal or be measured by adapted detection elements of the terminal and communicated to the transponder, for example, by initiating a specific communication during the training phase. Rather than storing the different values, the transponder may 
         [0000]    
       
         
           
             ( 
             
               
                 block 
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                 63 
               
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                         opt 
                       
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                     20 
                   
                 
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                         20 
                       
                       
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         [0000]    directly calculate value V C2opt]R20  by applying formula 10 and store the result. 
         [0108]    Various embodiments with different variations have been described hereabove. It should be noted that those skilled in the art can combine various elements of these various embodiments and variations without showing any inventive step. In particular, the selection and the order of the tests to be performed depend on the application, for example, on the time available to perform the authentication, on the calculating capacity of the transponder, etc. 
         [0109]    Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and the scope of the present invention. Accordingly, the foregoing description is by way of example only and is not intended to be limiting. The present invention is limited only as defined in the following claims and the equivalents thereto.