Patent Publication Number: US-10318245-B2

Title: Device and method for determining an inverse of a value related to a modulus

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This Application is a Continuation of U.S. application Ser. No. 11/871,314, filed on Oct. 12, 2007. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to cryptographic algorithms and, in particular, to algorithms for determining an inverse of a value related to a modulus. 
     Computations of inverses, in particular, a computation of the modular multiplicative inverse Z 0   −1  mod N 0 , wherein Z 0  and N 0  are two non-negative integers with gcd(Z 0 ,N 0 )=1 (gcd= g reatest  c ommon  d ivisor) are e.g. an important component when determining cryptographic keys. They are needed e.g. when determining cryptographic keys for the RSA-algorithm (named after Ron Rivest, Adi Shamir and Leonard Adleman) or other cryptographic algorithms as e.g. the ECDSA-algorithm (ECDSA=Elliptic Curve Digital Signature Algorithm). In this context, the integer Z 0  or N 0  is a secret that should not be revealed by an attacker. 
     The common algorithm used for computing Z 0   −1  mod N 0  is the extended Euclidean algorithm which results from the Euclidean algorithm that is extended so that it not only yields the greatest common divisor of two integers Z 0  and N 0 , but also integers x and y satisfying Z 0 x+N 0 y=d, where d=gcd(Z 0 ,N 0 ). 
     The extended Euclidean algorithm contains a division in an iteration loop (e.g. a while-loop) wherein the division is again typically realized by another iteration loop in which the involved integers are shifted and subtracted or added. 
     There are several methods to restructure the extended Euclidean algorithm to realize it on a microprocessor. Typically, all these methods or variants have one thing in common: They consist of an outer and an inner iteration loop, wherein the outer iteration loop corresponds to a loop exchanging integer pairs and the inner loop corresponds to the implementation of the division. Routines of this kind are susceptible to SPA (SPA=Simple Power Analysis) attacks since the current or power consumption and, additionally, the time consumption depend on the numbers to be processed. An attacker could thus draw conclusions as to the numbers processed from the current or time profile and thus for example spy out a secret key of a public-key crypto algorithm as e.g. the input Z 0 . 
     Hence, it is desirable to implement the division within the outer iteration loop more securely. 
     SUMMARY OF THE INVENTION 
     Embodiments of the present invention provide a device for determining an inverse of an initial value related to a modulus comprising a unit for processing an iterative algorithm in a plurality of iterations, wherein an iteration includes two modular reductions and has, as an iteration loop result, values obtained by an iteration loop of the extended Euclidean algorithm. 
     Further, embodiments of the present invention provide a device for determining an inverse of an initial value related to a modulus comprising a unit for processing an iterative algorithm in a plurality of iterations, the unit being adapted for performing, within an iteration, the following steps: determining a preliminary first iteration result value based on the first iteration result value of a preceding iteration or an initial first iteration result value of a first iteration and a modular reduction with respect to a modulus determined using an iteration modulus of a preceding iteration or the initial value in case of a first iteration loop, a second iteration result value of the preceding iteration or an initial second iteration result value of a first iteration and an enlargement parameter, performing a modular reduction of an iteration value with respect to the iteration modulus resulting in an updated iteration value, and extracting a first iteration result value from the preliminary first iteration result value using the updated iteration value. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Preferred embodiments of the present invention will be described in detail referring to the accompanying drawings, in which: 
         FIGS. 1 a  and 1 b    show flow charts of the extended Euclidean algorithm; 
         FIG. 2  shows a block diagram of a device for determining an inverse according to an embodiment of the present invention; and 
         FIGS. 3 a  and 3 b    show flow charts of an algorithm for determining an inverse according to embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1 a    shows a flow chart of the conventional extended Euclidean algorithm. 
     Inputs into the extended Euclidean algorithm are two non-negative integers N 0 , Z 0  with gcd(Z 0 ,N 0 )=1. The output of the extended Euclidean algorithm is then the modular multiplicative inverse Z 0  mod N 0 . 
     In a first step S 10 , initial first and second iteration result values n 0 , z 0  are determined, respectively. The initial first iteration result value n 0  is set to 0. The initial second iteration result value z 0  is set to 1 in step S 10 , as well as the iteration counter i. 
     In a second step S 20  of the extended Euclidean algorithm, an iteration loop is executed as long as an iteration loop requirement, i.e. an iteration modulus Z i−1  is greater than 0, is fulfilled. Thereby, the iteration modulus Z i−1  corresponds to the iteration modulus determined in the preceding iteration to the i-th iteration or the initial value Z 0  in case of the first iteration, i.e. i=1. 
     Within the i-th iteration of the iteration loop S 20  an integer result 
                     q   i     =     ⌊       N     i   -   1         Z     i   -   1         ⌋             (   1   )               
of a division of a numerator N i−1  by a denominator in form of the iteration modulus Z i−1  is computed in step S 21 . In a next iteration step S 22  within the i-th iteration, a value N i  is computed according to
 
 N   i   =N   i−1   −q   i   Z   i−1 ,  (2)
 
wherein N i  corresponds to the remainder of the integer division according to equation (1) and shall be denoted as iteration value in the following. N i−1  shall be denoted as iteration value of the preceding, i.e. the (i−1)-th iteration. N i  could also be computed according to
 
 N   i   =N   i−1  mod  Z   i−1 ,  (3)
 
     In a further iteration step S 23  of the extended Euclidean algorithm a first iteration result value n i  is determined according to
 
 n   i   =n   i−1   −q   i   ·Z   i−1 ,  (4)
 
     wherein n i−1  denotes the first iteration result value of the preceding iteration or the initial first iteration result value n 0  in case i=1. 
     In a next iteration step S 24 , the iteration modulus Z i  of the current, i.e. i-th, iteration is set to the iteration modulus Z i−1  of the preceding, i.e. (i−1)-th, iteration. Likewise the second iteration result value z i  of the current iteration is set to the second iteration result value of the preceding, i.e. (i−1)-th, iteration. Further, the first iteration result value n i  of the i-th iteration is exchanged or swapped with the second iteration result value z i  of the current iteration such that z i  becomes n i  and n i  becomes z i . Further, the i-th iteration value N i  is exchanged or swapped with the i-th iteration modulus Z i  such that Z i  becomes N i  and N i  becomes Z i . For swapping two values, an auxiliary register is typically required for temporarily storing one of the two values to be swapped. 
     In a next step S 25  the iteration counter i is incremented by one. 
     As long as the iteration loop requirement, i.e. Z i−1 &gt;0, is fulfilled, the iteration loop S 20  is performed. If the iteration loop requirement does not hold, i.e. Z i−1 ≤0, the iteration loop S 20  is left and the first iteration result value n i−1  of the last executed iteration which contains the desired multiplicative inverse Z 0   −1  mod N 0  is returned in step S 30  of the extended Euclidean algorithm depicted in  FIG. 1   a.    
     For a person skilled in the art it is obvious that the indexing depicted in  FIG. 1 a    could also look different. Furthermore, the swapping of the register contents in step S 24  can be omitted if the method is performed as depicted in  FIG. 1 b   . The results are, of course, identical, respectively. 
     As can be seen, the extended Euclidean algorithm contains an integer division in iteration step S 21  of the iteration loop S 20 . Typically, this integer division is again realized by a loop in which the involved numbers or respective register contents (here: N i−1 , Z i−1 ) get shifted and subtracted or added. It can be seen from iteration step S 23 , that the multiplicative inverse Z 0   −1  mod N 0  is iteratively determined in n i  using the integer result q i . With e.g. a simple power analysis (SPA) an attacker might be able to find out the secret initial value Z 0  by running the complete computation in reverse. 
     By trying to securely implement the division step S 21 , it might be possible to hide the exact value of the integer result q i . This can be achieved by designing decisions during program or algorithm execution such that it is not possible to find out which program path is currently taken. However, by doing this, there might still be enough side channel information to expose valuable information to an attacker. 
     Embodiments of the present invention solve this problem by replacing the integer division using a modular reduction being available for almost every cryptographic coprocessor. That is, a modular reduction takes the role of the division in embodiments of the present invention. Hence, the integer result q i  does not appear directly. 
       FIG. 2  shows a device  100  for determining an inverse of the initial value Z 0  related to the initial modulus N 0  according to an embodiment of the present invention. 
     The device  100  comprises a unit  110  for processing an iterative algorithm in a plurality of iterations, wherein an iteration includes two modular reductions and has, as an iteration loop result, values obtained by a loop of the extended Euclidean algorithm. 
     According to embodiments of the present invention, the unit  110  is adapted for performing an iterative algorithm and comprises a first register  112  for the initial value Z 0  being also used for an iteration modulus value Z i . Further, the unit  110  comprises a second register  114  for the initial modulus N 0  also being used for an iteration value N i . The unit  110  also comprises a third register  116  for an initial first iteration result value n 0  also being used for a first iteration result value n i  and a preliminary first iteration result value n i ′. A fourth register  118  is comprised for an initial second iteration result value z 0  being also used for an second iteration result value z i . Further, the unit  110  comprises a fifth register  120  for an enlargement parameter t i , a sixth auxiliary register  122  for a swapping operation, and a processing sub-unit  130  for accessing the registers to retrieve register contents when processing the iterative algorithm in the plurality of iterations. 
     In a further embodiment, the unit  110  comprises a seventh register  124  for storing a bit length l of the initial modulus N 0 . 
     An embodiment of the iterative algorithm shall be described in further detail referring to  FIG. 3   a.    
     According to an embodiment of the present invention, the unit  110  is adapted for setting initial values for a first n 0  and second iteration result value z 0 , and for setting a bit length l equating the bit length of the initial modulus N 0  in a first step S 40  before an iteration loop, i.e. n 0 =0, z 0 =1 and l=l(N 0 ). The iteration counter i is set to one. 
     In a next step S 50 , the plurality of iterations is performed as long as an iteration loop requirement is fulfilled. As for the extended Euclidean algorithm having been explained referring to  FIG. 1 , the plurality of iterations are executed as long as an iteration modulus Z i−1  is larger than 0. Step S 50  comprises iteration steps of the i-th iteration loop, with i being an integer greater than or equal 1. 
     The unit  110  is, according to an embodiment of the present invention, adapted for determining an enlargement parameter t i  in a step S 51 . Thereby, the enlargement parameter t i  depends on a bit length of the iteration modulus Z i−1  of the preceding iteration and on the bit length of the initial modulus N 0 . According to an embodiment of the present invention, the enlargement factor t i  is determined according to
 
 t   i ≥2 l+2−l(Z     i−1     ) ,  (5)
 
wherein l denotes the bit length of the initial modulus N 0  and l(Z i−1 ) denotes the bit length of the iteration modulus Z i−1  of the preceding iteration (i−1). In equation (5) the number 2 is the base since exemplarily only a binary number system is considered here, while the enlargement parameter t i  results when the base  2  is raised to the power of an exponent e. In general, the enlargement parameter t i  has to fulfill the following condition:
 
 t   i &gt;( q   i +1)| z   i−1 |  (6)
 
     The reason for the inequality (6) will become evident further below. 
     It is preferred to rather select the enlargement parameter to be small since the enlargement parameter determines the length of the registers required for calculating the division result. If the enlargement parameter was selected to be very large, very long registers would be required, while shorter registers are sufficient when the enlargement parameter t i  is selected to be smaller. Equation (5) indicates the preferred dimensioning of the exponent e for the binary case, i.e. e=l+2−l(Z i−1 ). 
     In a further iteration step S 52 , the unit  110  determines a preliminary first iteration result value n i ′ based on the first iteration result value n i−1  of a preceding iteration or the initial first iteration value n 0  in case of the first iteration and a modular reduction with respect to a modulus determined using the iteration modulus Z i−1  of a preceding iteration or the initial value Z 0  in the case of the first iteration, and the second iteration result value z i−1  of the preceding iteration or the initial second iteration result value z 0  in case of the first iteration, and the enlargement parameter t i . In particular, the unit  110  determines the preliminary first iteration result value n i ′ in step S 52  based on the following equation
 
 n   i   ′=n   i−1 +[( N   i−1   ·t   i )mod( z   i−1   +Z   i−1   ·t   i )],  (7)
 
wherein n i ′ denotes the preliminary first iteration result value, n i-1  denotes the first iteration result value of the preceding iteration or the initial first iteration result value n 0  for i=1, N i−1  denotes the iteration value of the preceding iteration or the initial modulus N 0  for i=1, z i−1  denotes the second iteration result value of the preceding iteration or the initial second iteration result value z 0  for i=1, Z i−1  denotes the iteration modulus of the preceding iteration or the initial value Z 0  in case of the first iteration (i=1) and t i  denotes the enlargement parameter of the i-th iteration.
 
     A multiplication with the enlargement parameter t i  can be efficiently realized by a shift operation to shift the respective register contents for N i−1  or Z i−1  to more significant bit positions, wherein the number of shifted bit positions corresponds to the exponent e=l+2−l(Z i−1 ) of the base  2 . Hence, the enlargement parameter t i  can be regarded as a modulus shift value and (Z i−1 t i ) can be regarded as an enlarged modulus. 
     The modulus shift value is, according to embodiments, determined by the processing sub-unit or the controller  130 , wherein the controller  130  comprises a first processing sub-unit  140 , e.g. a general purpose microprocessor, for processing numbers having a first bit length, e.g. 64 bits. Further, the unit  110  comprises a second processing sub-unit  150 , e.g. a cryptographic coprocessor, for processing numbers having a second wordlength, e.g. 1024 bits, the second wordlength being greater than the first bit length. Further, the controller  130  is configured to shift the content of the modulus register  112  to more significant bit positions by a number of register bit-positions e.g. corresponding to the exponent e=l+2−l(Z i−1 ) to obtain a shifted content. 
     In a next iteration step S 53 , a modular reduction of the iteration value N i−1  of the preceding iteration with respect to the iteration modulus Z i−1  of the preceding iteration is performed according to
 
 N   i   =N   i−1  mod  Z   i−1   (8)
 
resulting in an updated iteration value N i .
 
     In yet a further iteration step S 54 , the first iteration result value n i  is extracted from the preliminary first iteration result value n i ′ using the iteration value N i  according to
 
 n   i   =n   i   ′−N   i   ·t   i .  (9)
 
     In a last iteration step S 55  of an embodiment of the present invention, the iteration modulus Z i  of the current, i.e. i-th, iteration is set to the iteration modulus Z i−1  of the preceding, i.e. (i−1)-th, iteration. Likewise the second iteration result value z i  of the current iteration is set to the second iteration result value z i−1  of the preceding, i.e. (i−1)-th, iteration. Further, values between the first and second iteration result values n i , z i  and between the iteration modulus Z i  and the iteration value N i  are swapped or exchanged such that z i  becomes n i  and n i  becomes z i  and such that Z i  becomes N i  and N i  becomes Z i . 
     In other words, the contents of the register  116  of the device  100  for the first iteration result value n i  or the initial first iteration result value n 0  in case of a first iteration and the register  118  of the device  100  for the second iteration result value z i , the second iteration result value of the preceding iteration z i−1  or the initial second iteration result value z 0  in case of a first iteration are swapped. Further, the contents of the register  112  of the device  100  for the iteration modulus Z i , the iteration modulus of the preceding iteration Z i−1  or the initial value Z 0  in case of a first iteration and the register  114  of the device  100  for the updated iteration value N i , the iteration value of the preceding iteration N i−1  or the initial modulus N 0  in case of a first iteration are swapped. 
     In further embodiments of the present invention the step S 55  of swapping can also be performed at the beginning of each iteration by swapping register contents for the first iteration result value and the second iteration result value of the preceding iteration and by swapping register contents for the iteration modulus and the iteration value of the preceding iteration, or the respective initial values in case of a first iteration. 
     In a next step S 56  the iteration counter i is incremented by one. 
     Before the next iteration of the iteration loop S 50 , the iteration requirement is checked, i.e. it is checked whether the iteration modulus Z i−1  of the preceding iteration is greater than 0. 
     If this is the case, the next iteration is executed. If the iteration requirement is not fulfilled, then the first iteration result value n i−1  of the last executed iteration is returned in a step S 60  containing Z 0   −1  mod N 0  or (Z 0   −1  mod N 0 )−N 0 . 
     Further, the swapping operation in step S 55  could also be omitted by simple algorithmic modifications. If, e.g., n i  was replaced by z i ′ in step S 52 , N i  was replaced by Z i  in step S 53 , n i , n i ′ and N i  were replaced by z i , z i ′ and Z i , respectively, in step S 54 , and Z i =Z i−1 , z i =z i−1  were replaced by N i =N i−1 , n i =n i−1  respectively, in step S 55 , the swapping operation in step S 55  could be omitted as depicted in  FIG. 3   b.    
     For a person skilled in the art it is further obvious that the indexing depicted in  FIG. 3 a    could also look different. 
     The cryptographic coprocessor  150  is adapted to perform the modular reduction according to equation (7) and/or (8). According to embodiments of the present invention in the modular reduction algorithm a modular reduction is performed on a target value by subtracting an enlarged modulus. For that, not necessarily in the cryptographic coprocessor  150 , a modulus shift value is determined and a modulus is enlarged using the modulus shift value to obtain the enlarged modulus. The enlarged modulus is then subtracted from the target value. 
     The general purpose microprocessor  140  performs an iteration control of the iterative algorithm or the iteration-loop S 50  according to embodiments of the present invention. 
     In the following the inventive algorithm depicted in the flow chart of  FIG. 3 a    shall be described in more detail, and compared to the extended Euclidean algorithm depicted in the flow chart of  FIG. 1   a.    
     Using equation (2) of step S 22  of the extended Euclidean algorithm, equation (1) of step S 21  can be transformed as follows:
 
 N   i−1   =q   i   ·Z   i−1   +N   i ,  (10)
 
     wherein the value N i  is greater than or equal to 0 and smaller than Z i−1 . When equation (10) is multiplied by the enlargement parameter t i , the following equation results:
 
 N   i−1   ·t   1   =q   i   Z   i−1   ·t   i   +N   i   ·t   i .  (11)
 
     When additionally equation (3) is also multiplied by the enlargement parameter t i  on both sides, the following equation results:
 
 N   i   ·t   i   =N   i−1   ·t   i  mod( Z   i−1   ·t   i ).  (12)
 
     In addition, the following applies:
 
0 ≤N   i   ·t   i   &lt;Z   i−1   ·t   i .  (13)
 
     The z i−1 -fold integer result q i  is then added to (or subtracted from), and simultaneously subtracted from (or added to) the right side of equation (11), which corresponds to the following equation:
 
 N   i−1   ·t   i   =q   i   Z   i−1   ·t   i   +z   i−1   q   i   +N   i   ·t   i   −z   i−1   q   i .  (14)
 
     When equation (14) is transformed, such that the result q i  of the first two terms on the right hand side of equation (14) is factored out, the following expression results:
 
 N   i−1   t   i   =q   1 ( Z   i−1   ·t   i   +z   i−1 )+ N   i   ·t   i   −z   i−1   q   i .  (15)
 
     Equation (15) can be transformed to
 
 N   i   ·t   i   −z   i−1   q   i   =N   i−1   t   i   −q   i ( Z   i−1   ·t   i   +z   i− )  (16)
 
     When equation (16) is then compared to equations (10) and (2), it becomes evident that equation (16) is a new determination equation for a new division, wherein the difference for the sum of the left side of equation (16), i.e. the auxiliary quantity (N i t i −z i−1 q i ) in which the result q i  sought for is contained, corresponds to the remainder of an integer division of a numerator N i−1 t i  by a denominator (Z i−1 t i +z i−1 ). The remainder of this division, i.e. the auxiliary quantity on the left side of equation (16) can be calculated by the following equation in analogy to equation (3):
 
 N   i   ·t   i   −z   i−1   q   i =( N   i−1   t   i )mod( Z   i−1   ·t   i   +z   i−1 )  (17)
 
     When equation (17) with the difference on the left side is inserted into equation (7), the following results:
 
 n   i   ′=n   i−1   +N   i   ·t   i   −z   i−1   q   i .  (18)
 
     When equation (18) is then inserted into equation (9) the following equation results:
 
 n   i   ′=n   i−1   −z   i−1   q   i .  (19)
 
     Hence, equation (19) corresponds to equation (4) of step S 23  of the extended Euclidean algorithm to determine the multiplicative inverse Z 0   −1  mod N 0 . 
     Looking at equation (17) and coming back to the inequality (6) it becomes obvious to those skilled in the art why the enlargement parameter t i  should be, in any case, greater than (q i +1)|z i−1 |. This is to ensure that the left-hand side of equation (17) does not become negative at any time. In particular, the enlargement parameter t i  has to be big enough, such that the left-hand side lies between zero and the modulus of the right-hand side of equation (17). 
     According to embodiments of the present invention the device  100  is hence configured as a side-channel-attack-secure-reduction hardware block and is at least part of a key generator for keys used in cryptographic applications. The device  100  is thereby adapted for generating e.g. a private key for an asymmetric cryptographic scheme, the cryptographic scheme comprising a key pair of the private key and a corresponding public key. The private key is kept secret, while the public key may be widely distributed. The keys are related mathematically, but the private key cannot be practically derived from the public key. A message encrypted with the public key can be decrypted only with the corresponding private key. For that the device  100  outputs a value (the modular multiplicative inverse) determined in a last iteration of a plurality of iterations as at least a part of the private key. 
     Hence, the present invention, due to its flexibility, safety and performance, is suitable in particular for cryptographic algorithms and for cryptographic coprocessors on which a safe and efficient implementation of the modular reduction is typically implemented by means of a circuit, in particular for cryptographic key-generation. 
     Depending on the circumstances, the inventive method may be implemented in hardware or in software. The implementation may be done on a digital storage medium, particularly a disk or a CD with electronically readable control signals, which may cooperate with a programmable computer system so that the method is executed. In general, the invention thus also consists in a computer program product with a program code stored on a machine-readable carrier for performing the inventive method when the computer program product runs on a computer. In other words, the invention may thus be realized as a computer program with a program code for performing the method when the computer program runs on a computer. 
     While this invention has been described in terms of several preferred embodiments, there are alterations, permutations, and equivalents which fall within the scope of this invention. It should also be noted that there are many alternative ways of implementing the methods and compositions of the present invention. It is therefore intended that the following appended claims be interpreted as including all such alterations, permutations, and equivalents as fall within the true spirit and scope of the present invention.