Abstract:
The present invention relates to the field of computer data encrypting and decrypting, especially for mobile equipments like PDA, mobile phones, smart cards and the like, which need a good trade-off between computing speed, power consumption and security strength. Embodiments of the invention provide encrypting/decrypting methods implementing simple data operation. Such methods are based on generating a pseudo-random sequence through a function of the Collatz (or Syracuse) family from a starting number used as a secret key.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of co-pending European Patent Application No. EP07113715, filed 2 Aug. 2007, which is hereby incorporated herein. 
     FIELD OF THE INVENTION 
     The present invention relates to the field of computer data encrypting and decrypting, especially for compact or low performance or power devices such as smartcards or nomad and mobile computerized objects. 
     BACKGROUND OF THE INVENTION 
     In encryptography, more and more processing power is required to encipher or decipher texts or data. This often calls for dedicated “hardware assist” components which need substantial computer resources (memory, CPU cycles) which themselves ask for significant energy sources. On mobile equipment such as PDAs, mobile phones, smart cards and the like, it is desirable to have systems which present a good trade-off between power consumption and security strength. 
     SUMMARY OF THE INVENTION 
     In one embodiment, the invention comprises processing at least one pseudo-random sequence of numbers generated from at least one first key for encrypting or decrypting data. Generation of this pseudo-random sequence of numbers comprises an iteration of a function termed pseudo-random function, which is defined as comprising the following steps: testing a determined test condition on a first number from this sequence; in at least a first case of said test condition, applying on said first number a first operation; in at least a second case of this test condition, applying on this first number a second operation; using result of this first operation or second operation for obtaining a second number, this second number taking place in this sequence after this first number. 
     First and second operations are two different arithmetical functions. They are selected so as, for at least one of these two operations, when a first number is processed through such operation issuing a second number, the result of the test condition on the second number is not systematically identical to the result of the test condition on the first number. Preferably, both first and second operations are selected under such a condition. 
     As an example, if the test condition is a parity test, any function which may never cause a parity change cannot be chosen as such an operation. Thus, adding an even number or multiplying with an even number may not be selected as such an operation. This potential change of test condition between first and second number of the pseudo-random sequence is a factor for a more randomly distributed sequence. 
     Preferably, the invention proposes to generate the second number through a function of the Collatz type, as defined hereafter. In a preferred embodiment, the first number is an integer and the step of checking the test condition comprises calculating parity of this first number. Parity computing is quite simple and fast done in binary circuits, and enables good performance with low complexity and power consumption. 
     In alternative embodiments, the test condition may comprise calculating a value of this first number under a modular equality. As an example, the method according to the invention may involve three cases and three operations, depending on a test condition of equality modulo 3. 
     According to embodiments of the invention, the pseudo-random sequence of numbers is used for encrypting or decrypting binary data, through a method comprising the following steps: generating the pseudo-random sequence of numbers from a first key data, termed starting number, treated as an initial first number for this pseudo-random sequence of numbers; processing this pseudo-random sequence of numbers through a conversion treatment resulting into a pseudo-random of binary digits; applying a encyphering or decyphering treatment, using this binary pseudo-random sequence as a seed for encrypting or respectively decrypting computer data. 
     Preferably, first and second operations are chosen such that the result of applying the first operation on the first number is greater than this first number, while the result of applying the second operation on this same first number is lesser than this first number, or reversely. This feature enables the sequence to involve numbers staying relatively low, thus minimizing the need for large binary registers or memories. Also, it combines well with the conversion treatment described hereabove for issuing a more randomly distributed pseudo-random binary sequence. For a better device simplicity and an optimal trade-off between different technical constraints, as well as a better “random quality” or unpredictability of the pseudo-sequences generated, the invention proposes using functions with the following features, as first and/or second operations: applying the first operation on the first number comprises dividing this first number by a determined number greater than one; applying the second operation on the first number comprises multiplying this first number by another number greater than one, the result of which being further added with an odd number. 
     Furthermore, according to embodiments of the invention, the test condition and first and second operations involve the following features. The step of checking the test condition results in the first case when the first number parity is even. The step of applying the first operation to this first number then comprises dividing this first number by an even integer. Meanwhile, the step of checking the test condition results in the second case when the first number parity is odd. The step of applying the second operation to this first number then comprises multiplying this first number with another integer greater than one, the result of which being then added with one. 
     Also, the function is selected so as to ensure that the function cannot “loop on itself”, meaning that for any starting first number, the function will always, after multiple iterations, converge to the same fixed number. 
     Alternatively, the method moreover comprises a step of verifying that the function is not looping on itself, e.g. through verifying that the second number was not already obtained in the pseudo-random sequence of numbers. 
     In the preferred embodiment described hereafter, first and second operations are defined as follows. In the first case, i.e. when first number is even, the step of applying the first operation to the first number further comprises dividing this first number by two. In the second case, I.e. when first number is odd, the step of applying the second operation to the first number further comprises multiplying this first number with three, the result of which being then added with one. 
     According to the preferred embodiment, the step of encyphering binary data, termed plain data, into encrypted binary data furthermore comprises the following steps: splitting the plain data into a sequence of consecutive binary words, termed word sequence, of a length based on a second key data; generating a sequence of numbers, termed encrypted sequence, from this word sequence, where at least one binary word from this word sequence is replaced with an number representing at least one position containing this binary word within the pseudo-random binary sequence; generating this encrypted binary data from this encrypted sequence. 
     In the reverse way, the step of decyphering encrypted binary data into decrypted binary data furthermore comprises the following steps: reading this encrypted data into a sequence of numbers, termed encrypted sequence; generating a sequence of binary data words, termed word sequence, from this encrypted sequence, where at least one number of this encrypted sequence is used as an offset for reading, whithin the pseudo-random binary sequence, a binary word the length of which is based on a second key data, this number of this encrypted sequence being replaced with this binary word into this word sequence; concatenating this word sequence into decrypted data. 
     A computerized device or system is also provided in embodiments of the invention, implementing such encrypting and/or decrypting method into software processing processor, or hardware or mixed circuits. Embodiments of the invention also provide a computer program the instructions of which carry out the steps of such a method, when this computer program is executed on a computer system. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The new and inventive features believed characteristic of the invention are set forth in the appended claims. The invention itself, however, as well as a preferred mode of use, further objects and advantages thereof, will best be understood by reference to the following detailed description of an illustrative detailed embodiment when read in conjunction with the accompanying drawings, wherein: 
         FIG. 1  schematically illustrates the progress of an encrypting process according to the invention; 
         FIG. 2  schematically illustrates the progress of a decrypting process according to the invention; 
         FIG. 3  is a block diagram illustrating an encrypted data transmission method between an emitter and a receiver, according to the invention; 
         FIG. 4  is a diagram illustrating the pseudo-random sequence of numbers for an example starting number value of 27, according to the preferred embodiment of the invention; 
         FIG. 5  is a table showing the distribution pattern of available offsets, for all possible word values with an example word length value of 4 bits, for the 64 first starting numbers which enable all such values, according to the preferred embodiment of the invention; 
         FIG. 6  is a table showing the distribution pattern of available offsets, for a binary word with an example value of 14, among the 64 first starting numbers which enable all values of binary words with an example length of 4 bits; 
         FIG. 7  is an histogram showing the distribution of the number of possible starting numbers for ciphering the example binary word of  FIG. 6 , among the same 64 first starting numbers. 
     
    
    
     In the following specifications, elements common to several figures are referenced through a common identifier. 
     DETAILED DESCRIPTION OF THE INVENTION 
     A preferred embodiment of the invention is based on a pseudo-random sequence generated by a function of a Collatz type. 
     Collatz Functions 
     The original Collatz function is defined as follows: 
     Consider the following operation on an arbitrary positive integer: 
     If the number is even, divide it by two. 
     If the number is odd, triple it and add one. 
     For example, if this operation is performed on 3, the result is 10; if it is performed on 28, the result is 14. There is an unsolved conjecture in mathematics, based on this function, called the Collatz conjecture. It is named after Lothar Collatz, who first proposed it in 1937. This conjecture is also known as the “3n+1” conjecture, the Ulam conjecture (after Stanislaw Ulam), or the Syracuse problem. This conjecture asks whether a sequence based on the Collatz function, or a certain kind of number sequence, always ends in the same way regardless of the starting number. Paul Erdos said about the Collatz conjecture: “Mathematics is not yet ready for such problems.” He offered $500 for its solution. 
     In mathematical notation, we can define the Syracuse (or Collatz) function “S” in its original form as follows: 
     
       
         
           
             
               S 
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         n 
                         2 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       ≡ 
                       
                         0 
                         ⁡ 
                         
                           [ 
                           2 
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           3 
                           × 
                           n 
                         
                         + 
                         1 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       ≡ 
                       
                         1 
                         ⁡ 
                         
                           [ 
                           2 
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Starting with an initial number S 0 , it is possible to generate the sequence of “Syracused Numbers” as defined below, until the value 1 is reached:
 
 SN   0   =S   0  
 
 SN   i+1   =S ( SN   i )
 
     In the Syracuse conjecture literature, the following jargon is usually adopted:
         This sequence {SN i } is known as the flight of S.   Each SN i  is a stage of the flight.   The highest SN i  is known as the maximal elevation of the flight.   The duration of the flight is the number of stages before reaching the value 1.   The flight in elevation is the number of stages before going under the initial value S 0 .       

     The expansion factor is the ratio between the maximal elevation and the starting value S 0 . 
     Some examples of sequence characteristics for the original Syracuse/Collatz function: 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
               
                   
                   
                 Flight 
                   
                   
               
               
                   
                   
                 in 
                   
               
               
                   
                   
                 ele- 
                   
               
               
                   
                 dura- 
                 va- 
                   
                 Expansion 
               
               
                 S 0   
                 tion 
                 tion 
                 Maximal elevation 
                 factor 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 16 
                 11 
                 52 
                 7.43 
               
               
                 32 
                 5 
                 1 
                 32 
                 1 
               
               
                 27 
                 111 
                 96 
                 9232 
                 341.93 
               
               
                 97 
                 118 
                 3 
                 9232 
                 95.18 
               
               
                 2 50  + 1 
                 332 
                 3 
                 3377699720527876 
                 3 
               
               
                 871 
                 178 
                 57 
                 190996 
                 219.28 
               
               
                 703 
                 170 
                 132 
                 250504 
                 356.34 
               
               
                 100759293214567 
                 1820 
                 166 
                 1180174841128253392 
                 11712.81 
               
               
                   
               
             
          
         
       
     
     This original function may be generalized into a type of functions called Collatz type. A function G is called an Collatz type function if there is an integer n together with rational numbers {a i : i&lt;n}, {b i : i&lt;n} such that: 
     whenever x≅i mod p 
     then G(x)=a i x+b i  is integral. 
     The method according to the invention uses a function of Collatz type for generating the pseudo-random sequence of numbers. In a preferred embodiment described hereafter, the following Collatz type function is chosen for generating a pseudo-random sequence of numbers. 
     
       
         
           
             
               s 
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         n 
                         2 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       ≡ 
                       
                         0 
                         ⁡ 
                         
                           [ 
                           2 
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             3 
                             × 
                             n 
                           
                           + 
                           1 
                         
                         2 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       ≡ 
                       
                         1 
                         ⁡ 
                         
                           [ 
                           2 
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Some examples of sequence characteristics for this modified Syracuse/Collatz function, as used in the preferred embodiment described hereafter: 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
               
                   
                   
                 Flight 
                   
                   
               
               
                 S 0   
                 duration 
                 in elevation 
                 Maximal elevation 
                 Expansion factor 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 11 
                 6 
                 26 
                 3.71 
               
               
                 32 
                 5 
                 1 
                 32 
                 1 
               
               
                 27 
                 70 
                 59 
                 4616 
                 170.96 
               
               
                 97 
                 75 
                 1 
                 4616 
                 47.59 
               
               
                 871 
                 113 
                 34 
                 95498 
                 109.64 
               
               
                 703 
                 108 
                 80 
                 125252 
                 178.17 
               
               
                   
               
             
          
         
       
     
     Applicant assumes that the Syracuse conjecture is true. However, even in the opposite case, such functions nevertheless provide various pseudo-random sequences that are sufficiently numerous for building an encrypting/decrypting method with a good trade-off between security and power or speed performances.  FIG. 1  and  FIG. 2  respectively illustrate encrypting and decrypting of binary data according to the invention. In  FIG. 1 , a starting number S 0    110  is used as a secret key for encrypting plain binary data  114  comprising a sequence {bi} of binary bits. This starting number  110  is used as an initial first number for generating  121  and memorizing a pseudo-random sequence of numbers {si}  112 , through iteration of the pseudo-random function. The generated pseudo-random sequence  112  of numbers is then processed through a conversion treatment  122 , resulting into a pseudo-random sequence  113  of binary digits {sbi}. 
     Preferably, the conversion treatment  122  comprises the following steps:
         if said second number is greater than said first number, adding to the binary pseudo-random sequence a binary digit of a type, e.g. a bit with value “one”; or   if said second number is lesser than said first number, adding to the binary pseudo-random sequence a binary digit of the other type, e.g. a bit with value “zero”.       

     The resulting binary pseudo-random binary sequence  113  is then used as a seed for encyphering a sequence {bi} of binary data  114 , termed plain data, into a encrypted sequence {cbi} of binary data  117 . 
     This encyphering process comprises the following steps. Plain data  114  is converted  123  into a sequence  115  of consecutive binary words, termed word sequence{wi}, these words being of a length L based on a second key data  111 . This second key data may be used as a second secret key, possibly transmitted or detained separately from a first secret key based on the starting number  110 . The first  110  and second  111  key data may also be united or combined to form a unique secret key, which then need to be separated before use. 
     From this word sequence  115 , a encrypted sequence  116  of numbers {ni} is generated  124  through replacing each binary word w i  with a number n i  representing one position containing said binary word within the pseudo-random binary sequence  113 . The encrypted sequence of number  116  is then converted  125  into a sequence {cbi} of binary data  117 , providing the encrypted data  117  issued from the initial plain data  114 . 
     In  FIG. 2 , a starting number S 0    210  is used as a secret key for decrypting a encrypted binary data  214  comprising a sequence {cbi} of binary bits. In a manner that may be the same as in  FIG. 1 , a pseudo-random binary sequence  213  is generated  221 ,  222  from the same starting number  210 , which was once used for producing this encrypted binary data  214 . The resulting binary pseudo-random binary sequence  213  is then used for decyphering a sequence {cbi} of binary data  214 , termed encrypted data, into a plain sequence {bi} of binary data  217 . 
     The decyphering process comprises comes as follows. The encrypted data  214  is read  223  into a sequence of numbers {ni}, termed encrypted sequence  215 . A sequence of binary data words {wi}, termed word sequence  216  is generated  224  from the encrypted binary sequence  213 . Each number from this encrypted sequence of numbers  215  is used as an offset for selecting a reading position within the pseudo-random binary sequence  213 . Starting from this reading position, a binary word is read of a length L corresponding to the same second key data  211 , which was once used for producing this encrypted binary data  214 . 
     All the binary words of the resulting word sequence  216  are then concatenated  225  into a sequence of binary data {bi}, termed decrypted data  217 , which is then identical to the binary data that was once used for producing the encrypted binary data  214 . Although such ciphering and deciphering algorithm provides a good optimization when combined with pseudo-random sequences defined above, different algorithms may also be used for ciphering and deciphering plain data based on using such a pseudo-random binary sequence. 
       FIG. 3  illustrates more specifically a transmitting process of binary data  300  between an emitting device  301  and a receiving device  302 . Two parties “A”  301  and “B”  302  need to exchange a binary information  300  in a secret way. The following assumptions are made and the following notations are used in the rest of this example:
         Both parties A and B know  309  a secret key S 0 .   Both parties A and B know  309  a secret length L.   The binary information  300  to be shared from A to B is represented by a sequences of N bits {b i } i=1   i=N .   N is a multiple of L       

     The proposed method for ciphering the binary information is based on the following steps: In an initialisation stage  307 , both parties A  301  and B  302  build ( 312 , respectively  322 ) build the binary pseudo-random sequence {s i }defined by: 
     
       
         
           
             
               s 
               i 
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             s 
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                         &gt; 
                         
                           s 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             s 
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                         
                         &lt; 
                         
                           s 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     In its binary form {s i }, this sequence specifies the behavior of the Syracuse suite: does it go up (bit at “1”) or down (bit at “0”) at each successive step? 
     For each plain data  300  they wish to share, emission from A  301  to B  302  comprises the following steps: 
     In a processing stage  308 , the A party  301  splits  313  the plain text {b i } i=1   i=N    300  as a sequence of words {w j } j=1   j=N/L , defined as: 
     w j {b i } i=(j-1)×L+1   i=j×L . 
     For each word w j , the A party searches  314  in the sequence {s i } a series of L successive bits starting with offset n j  such that: w j {s i } i=n     j     i=n     j     +L−1 . If multiple solutions exist, the A party takes any of them in any way, possibly using a random or pseudo-random selection. 
     The A party sends  315  to the B party  302  the series {n j } i=1   =N/L  representing the genuine information {b i } i=1   i=N    300 , enciphered by the “Syracuse Secret Key” S 0 . 
     The B party  302  receives  323  the series {n j } i=1   =N/L  from the A party. 
     For each offset n j , the B party reconstructs  324  each word w j {s i } i=n     j     i=n     j     +L−1 . 
     From the sequence of words {w j } j=1   i=N/L , the B party reconstructs  325  the original information {b i } i=1   i=N    300 . 
     These steps can be implemented in various ways (hardware, software, hybrid), all following the logic described in the diagram of  FIG. 3 . 
     A person skilled in the art will easily understand that the proposed method and system asks for very few IT resources for its implementation. The required processing power is very low (simple operations like additions and shifts are needed), and the required memory is also very low (several bytes of ROM memory and few bytes of RAM memory are needed). 
     Example of a Data Transmission 
     The secret first key  110 ,  210  of a value S 0 =27 is secretly known by both parties A  301  and B  302 .  FIG. 4  shows the flight corresponding to the pseudo-random sequence  112 ,  212  generated for this value of “27” for the secret key. 
     The same pseudo-random binary sequence  113 ,  213  built by both parties A and B may be written as: 
     {s i }={1101111101011011101111010011101101111110011110001010100010011100001001}. 
     The secret second key  111 ,  211  of a value L=4 is known by both parties A and B, is used as a length for the words w i  of the word sequence  115 . 
     In this example, the genuine information  300 ,  114  that party A wants to transmit to party B under a encrypted form is defined as: 
     {b i } i=1   i=N ={1011111110101011011011011001101011011101}. 
     This genuine length N=40 is known by A. Thus, the party A splits  123  this information  300 ,  114  into a sequence  115  of ten words, each of 4 bits. Each word is then encrypted according to the pseudo-random binary sequence  113 . 
     The party A performs the following operations:
         w 1 ={1011}, so that n 1 ε{2, 10, 13, 17, 29, 32}; the value n 1 =17 is randomly selected.   w 2 ={1111}, so that n 2 ε{4, 5, 19, 34, 35, 36, 42}; the value n 2 =5 is randomly selected.   w 3 ={1010} so that n 3 ε{8, 22, 49, 51}; the value n 3 =8 is randomly selected.   w 4 ={1011} so that n 4 ε{2, 10, 13, 17, 29, 32}; the value n 4 =10 is randomly selected.   w 5 ={0110} so that n 5 ε{11, 30}; the value n 5 =11 is randomly selected.   w 6 ={1101} so that n 6 ε{1, 7, 12, 16, 21, 28, 31}; the value n 6 =16 is randomly selected.   w 7 ={1001} so that n 7 ε{24, 39, 57, 67}; the value n 7 =24 is randomly selected.   w 8 ={1010} so that n 8 ε{8, 22, 49, 51}; the value n 8 =8 is randomly selected.   w 9 ={1101} so that n 9 ε{1, 7, 12, 16, 21, 28, 31}; the value n 9 =31 is randomly selected.   w 10 ={1101} so that n 10 ε{1, 7, 12, 16, 21, 28, 31}; the value n 10 =12 is randomly selected.       

     Thus, the ciphered information {ni}  116  sent, e.g. under a standard binary form, from A to B is: 
     {n j } i=1   i=N/L ={17, 5, 8, 10, 11, 16, 24, 8, 31, 12}. 
     B party receives this sequence, e.g. under its binary form, and uses it as a sequence  215  of offsets for generating the plain binary data  217 . Thus, the party B applies each number of the encrypted sequence {n j } i=1   i=N/L    215  to the binary form  113 ,  213  of the pseudo random sequence {s i }  112 ,  212 , for deriving the sequence of words {w j } j=1   j=N/L    216 . Concatenation of the binary words from this word sequence  216  thus provides a binary sequence  217  identical to the genuine information  300 ,  114 : 
     {b i } i=1   i=N ={1011111110101011011011011001101011011101}. 
     Assume that a third party C wants to break the ciphered information, but ignoring both the secret key S 0  and the secret length L. This third party C assumes that the secret key is equal to 91 (wrong choice) and that the secret key is equal to 4 (right choice). Under these assumptions, we have for the party C:
         S 0 =91   {s i }={11011101111010011101101111110011110001010100010011100001001}
           the value n 1 =17 gives w 1 ={1101},   the value n 2 =5 gives w 2 ={1101},   the value n 3 =8 gives w 3 ={1111};   the value n 4 =10 gives w 4 ={1101};   the value n 5 =11 gives w 5 ={1010};   the value n 6 =16 gives w 6 ={1110};   the value n 7 =24 gives w 7 ={1111};   the value n 8 =8 gives w 8 ={1111};   the value n 9 =31 gives w 9 ={1111};   
           the value n 10 =12 gives w 10 ={0100}       

     The resulting deciphered information is: 
     {1 10 111 0 11 1 1 1 1 10 1 10 1011 10 1 11 11 1 1 1 11 1 1 0 10 0 } 
     where underscored digits are wrong (18 out of 40). 
     Thus it can be seen that the ciphered information  117 ,  214  is indeed a encrypted form af the genuine plain data  114 ,  217 . 
     According to selected combinations of length L and starting number S 0 , strength and flexibility of the encryption may vary. Flexibility must be sufficient for encryption of the genuine data intended to be transmitted, i.e. each binary word to be encrypted  115  must be found at least once under its binary form within the generated  122  binary pseudo-random sequence  113 . Furthermore, when only one offset exists for such a word, breaking the code may be easier than if several offsets are possible. 
       FIG. 5  to  FIG. 7  illustrates an example of distribution for the coding possibilities for a word length L of 4 bits. A 4 bits-word may takes 72 different values, ranging from {0000} to {1111}. 
     Within the pseudo-random binary sequence generated from an integer taken as starting number, it is not always possible to find an offset with every combination of such a 4 bits-word. The more long the word, the harder it becomes. Thus, only a part of the possible keys S 0  enable to code any value of such a word. Such keys may be termed “full keys”, for a given word length. 
       FIG. 5  is a table showing a distribution pattern of available offsets, for all possible word values with length value of 4 bits. This table shows the 64 first starting numbers which may be used as full keys for such a word. The top title line  501  shows the values of these 64 first full keys. All possible decimal value of a 4 bits binary word stand in the left title column  502 , while the total number of possible offset for each word value stands in the right column  503 . 
     For instance, starting number  27  results in a pseudo-random binary sequence which offers 7 different offsets corresponding to the word {1110}, i.e. with value  14 . Also, this word value  14  may be coded in 422 possibilities for the 64 first full keys. 
     It can be seen that numerous possibilities exist even for starting numbers quite low, thus enabling simple and compact computing or memorizing. 
     In the table of  FIG. 6 , cells in grey show the distribution pattern of these 422 available offsets n i  for the same 4-bits word value  14 . Offsets from 1 to 72 stand on the left title column  602 , while the starting numbers stand on the top title line  601 . For instance the offset pattern for the value SN 0 =27 (ref. 604 ) is equal to the set {6, 15, 20, 27, 37, 43, 60}. 
     On the right column  603  is reported, for each line, the number of starting numbers that may code this value  14  with the same offset. Thus, the value  14  coded at offset  6  (ref. 605 ) still leaves 8 (ref. 606 ) different possible keys among the 64 first full keys. These 8 possible keys are in the set {27, 82, 83, 103, 121, 194, 195, 233}. 
     In this specific example, it can be seen that different keys do not result in the same possible offsets, meaning that knowledge of the length and position of one specific word is usually not sufficient for retrieving the secret key. There are only a few similarities between different starting numbers. In this example, there are no more than 4 keys that have a similar distribution pattern (e.g. keys  193 ,  194 ,  195 ,  199 ). Also, all the possible offset values (on the left) are more or less equally visited, as seen in  FIG. 7 . 
     In  FIG. 7 , offsets from 1 to 72 stand on the bottom line  701 , while each bar of the histogram  702  shows the number of possible starting numbers for ciphering the same binary word of value  14 , among the same 64 first full keys. This example is one among several simulations that gave similar results, thus indicating an interesting encryption performance when balanced with the low need in power or speed resources. 
     In a preferred embodiment, selection of any starting number as a key may be validated through checking that this starting number is indeed a full key for the word length selected. 
     While the invention has been particularly shown and described mainly with reference to a preferred embodiment, it will be understood that various changes in form and detail may be made therein without departing from the spirit, and scope of the invention. In other embodiments, for example, possibly combined with the preferred one, starting numbers may be selected as keys even if not a full key. The encrypting method may then comprise a step of changing this key into another, through an algorithm shared between parties, e.g. by automatically selecting the next full key when encountering a word with no available offset in the initial key. Such a key modification may also be triggered on a test issuing a strength quality too low for the selected key, for some words or for all of them. Such a strength quality evaluation may be based on a low number  608  of possible keys for a given word at a given offset  607  of the pseudo-random binary sequence  113 ,  213 . 
     First and/or second operation may also be changed or modified, for the generation of the whole pseudo-random sequence or in the course of such a generation. Several pseudo-random sequences may also be used together, alternatively or interleaved.