Title of Invention  " A CRYPROGRAPHIC METHOD THAT SAVES COMPUTING OPERATIONS OR COMPUTATION TIME WHILE RETAINING OR INCREASING THE SECURITY AND APPARATUS THEREFOR". 

Abstract  The invention relates to a cryptographic method with at least one computing step containing a modular exponentiation E according to E = xd (mod poq), with a first prime factor p, a second prime factor q, an exponent d and a number x, whereby the modular exponentiation E is calculated according to the Chinese Remainder Theorem. 
Full Text   1  A CRYPTOGRAPHIC METHOD THAT SAVES COMPUTING OPERATIONS OR COMPUTATION TIME WHILE RETAINING OR INCREASING THE SECURITY AND APPARATUS THEREFOR. The present invention relates to a cryptocraphic method that saves computing operations or computation time while retaining or increasing the security and apparatus therefor. Cryptographic methods in the form of encryption and signature schemes are becoming increasingly widespread in particular due to the rising importance of electronic commerce. They are normally implemented by means of electronic apparatuses which may comprise for example a programmable universal microcontroller or else a specialized electronic circuit e.g. in the form of an ASIC. An especially interesting form of cryptographic apparatus is the smart card, since if expediently designed technically it can protect secret key data against unauthorized access. There are constant efforts both to improve the execution speed of cryptographic methods and to protect them against all possible kinds of attacks. The invention is suitable in particular for use in connection with smart cards, but is in no way limited thereto. It is instead implementable in connection with all kinds of cryptographic apparatuses. In a number of known cryptographic methods it is necessary to perform a modular exponentiation according to the equation where p and q are prime numbers. An especially important cryptographic method that comprises a modular exponentiation step is the RSA method, known for example from Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of Applied Cryptography," Boca Raton: CRC Press, 1997, pages 285 to 291. Use of modular exponentiation is not limited to the RSA method, however, but also includes for example Rabin signatures known from Menezes et al., ibid., pages 438 to 442, and the FiatShamir identification scheme known from Menezes et al., ibid., pages 408 to 410. The security of cryptographic methods that include modular exponentiation is regularly dependent on the difficulty of factorizing the number N from equation (1) into its prime factors p and q. This problem is only of sufficient complexity for large values N, SO that N should be chosen as great as possible, on the one hand. The computing effort for calculating values by means of modular exponentiation according to equation (I) increases monotonically with the magnitude of N, on the other hand, so 2 that it would be desirable from the point of view of practical applicability to be able to restrict the computation time required to acceptable values despite large values of TV. It is known to increase the computation speed by a factor of four by applying the socalled "Chinese Remainder Theorem," which permits for example larger values N at equal computation time. Instead of evaluating equation (1) directly, a transformation is performed according to (2) where (3) (4) One consequence of applying the Chinese Remainder Theorem is that the modular exponentiation is no longer done modulo N, i.e. modulo that number concealing its own prime factorization, but successively in a first partial step modulo p and a second partial step modulo q, i.e. knowledge of the prime factorization n = p.q to be kept secret is presupposed in this computing rule and leads to division of the total computing process into a first computing step (3) essentially involving the first prime factor, and a second computing step (4) essentially involving the second prime factor. The advantage is that exponent d in equation (1) must be defined modulo (p.q), whereas the exponents in equation (2) must only be defined modulo (p) or (q), where denotes the Eulerian function. Interestingly, an attack scheme on such cryptographic methods using modular exponentiation has recently become known by which the information on the prime factorization of Ncan be recovered from the faulty result of a disturbed modular exponentiation by suitable artificial intrusions in the otherwise undisturbed computing sequence, provided the concrete implementation makes use of the Chinese Remainder. Theorem according to equations (2) to (4). This attempt known as the "Bellcore attack" is described for example in Dan Boneh, Richard A. DeMillo and Richard J. Lioton: "On the importance of checking cryptographic protocols for faults," Advances in Cryptology  EUROCRYPT, 97. Lecture Notes in Computer Science 1233, Berlin: Sponger, 1997. An encryption device is manipulated by physical intrusions such as 3 increasing clock speed, operating voltage or irradiation so that computing errors occur in the execution of the modular exponentiation according to the Chinese Remainder Theorem with a certain, not too great probability. If a computing error occurs only in one of the two terms in equation (2), the two prime factors/p and q can be reconstructed from the erroneous exponentiation result. The consequence to be drawn from this vulnerability of modular exponentiation implemented by means of the Chinese Remainder Theorem is to first check the result of the computing operation for correctness before it is processed further, particularly before it is outputted in some form, e.g. in the form of a signature. A trivial countermeasure for the "Bellcore attack" is to effect this correctness check by repeating the computing operation at least once. In case of random computing errors it can be assumed that the result of the first computing operation deviates from that of the check computing operations. The essential disadvantage of this approach is that the computation time is already doubled by one check calculation. [ The print WOA1 9 8/52319 discloses in particular a method for protecting computing operations executing modular exponentiation according to the Chinese Remainder Theorem against the "Bellcore attack." A secret integer j is selected for example in the range of [0, 2kl] with 16 vl = x(modj.q) (5) v2 =x (mod j.q) (6) d1 = d (mod jop)) (7) d2 = d (mod (joq)) (8) w1 = v1d1 (mod jop) (9) w2 = v2 d2(mod joq) (10) Then it is checked whether it holds that: w1 = w2 (modj) (11) If expression (11) can be verified, the following expressions are calculated in the known method: y1= w1 (mod p) (12) 4 y2 = w2 (mod q) (13) from which the value for E = xd (mod N) (14) can then be determined by means of the Chinese Remainder Theorem. This known method has the advantage over simple check computing operations that the additional computation time required is substantially lower. In this method, both prime numbers p and q must be multiplied by the same factor d. The print WOA19 8/52319 describes a second method that permits prime numbers p and q to be multiplied by different factors r and s. However, two further exponentiations are possible for the check calculation. The problem of the invention is to provide a cryptographic method and apparatus that save computing operations or computation time while retaining or increasing the security. 4A Accordingly, the present invention provides a cryptographic method that saves computing operations or computation time while retaining or increasing the security, said method having at least one computing step containing a modular exponentiation E E = xd (mod poq) with a first prime factor p, a second prime factor q, an exponent d and a base x, whereby for carrying out the modular exponentiation two natural numbers r and s are chosen with the condition that d is relatively prime to kg V(r,s)), and whereby the following computing steps are performed: X1 = x (mod por) x2 = x (mod qos) d_2=d(mod pos)) d_2 = d (mod (qos)) Z1 =X1d2 (mod por) z2 = x2 d2 (mod qos) and whereby (.) is the Eulerian function and kgV(r.s) is the smallest common multiple of r and s, then a number z is calculated according to the Chinese Remainder Theorem fromz1 and z2 withz = z1 (modpor) ;z=z2(mod qos) ; the result E of the exponentiation is calculated by reduction of z modulo poq, the previously calculated number z and thus the result E is checked for computing errors in a checking step, the checking step comprises the following computing operations: calculating the smallest possible natural number e with the property eod=1 (mod kgV(r.s))) with the aid of the Extended Euclidean Algorithm, calculating the value C=ze (mod kgV(r,s)), comparing the values x and C modulo kgV(r,s), whereby the result of the modular exponentiation E is rejected as faulty if x ? C (mod kgV(r,s)). The present invention also provides a cryptographic method that saves computing operations or computation time while retaining or increasing the security, said method having at least one computing step containing a modular exponentiation E E = xu (mod poq) 4B with a first prime factor p, a second prime factor q, an exponent d and a base x, whereby for carrying out the modular exponentiation two natural numbers r and s, and two numbers b1 and b2 in the interval [I,...,r1] and [1,...s1] and relatively prime to r and s, respectively, are chosen, and whereby b1 and b2 fulfill the condition b1 = b2 (mod ggT(r,s)), where ggT(r,s) designates the greatest common divisor of r and s, the two numbers b1 and b2 are used to calculate according to the Chinese Remainder Theorem values x1 and x2 fulfilling the following conditions: X1 =x (mod p) ,x1 b1 (mod r) x2 = x (mod q), x2 = b2 (mod s) and then the following computing steps are performed: d_1 = d (mod (p)) d_2 = d (mod (q)) z2 =x1 d2 (mod por) Z2 = x2 d_2 (mod qos) and (.) represents the Eulerian function and kgV(r,s) the smallest common multiple of r and s, then a number z is calculated from z1 and z2 according to the Chinese Remainder Theorem with z = z1 (modpor) ;z=z2 (modqos) ; the result E of the exponentiation is calculated by reduction of z modulo poq, the previously calculated number z (and thus automatically also the result E) is checked for computing errors in a checking step, the checking step comprises the following computing operations: calculating the numbers C1 = b1 d_1 (modor) C2 = b2 d_2 (modos) whereby d_1 and d_2 are reduced before carrying out the modular exponentiation modulo r) and (s), respectively, comparing the values z1 and C1 modulo r as well as z2 and C2 modulo s, whereby the result of the modular exponentiation E is rejected as faulty if C, ? z1 mod r or C2 ? z2 mod s holds. The present invention further provides a cryptographic apparatus comprising : means for executing, with at least one exponentiation device, a computing step containing a modular exponentiation E E = xd (modpoq) with a first prime factory, a second prime factor q, an exponent d and a base x, whereby means for performing a modular exponentiation two natural numbers r and s are chosen with the condition that d is relatively prime to (kgV(r,s)), and whereby the following computing steps are performed: x1 =x (mod por) x2= x (mod qos) d_1 = d (mod d_2 = d (mod (qos)) z1 =x1 d1 (modpor) z2 = x2 d2 (mod qos) and (.) is the Eulerian function and kgV(r,s) is the smallest common multiple of r and s, means for calculating of the number z calculated from z1 and z2 according to the Chinese Remainder Theorem with z = Z1 (mod pr) z = z2 (mod qos); means for calculating the result E of the exponentiation by reduction of z modulo poq, e) means for checking the previously calculated number z (and thus automatically also the result E)for computing errors in a checking step, f) the checking step comprises the following computing operations: fl) calculating the smallest possible natural number e with the property eod=1 (mod kgV(r,s))) with the aid of the Extended Euclidean Algorithm, f2) calculating the value C = ze (mod kgV(r,s)), f3) comparing the values x and C modulo kgV(r,s), 'whereby the result of the modular exponentiation E is rejected as faulty if x ? C (mod kgV(r,s)). 4D The present invention further provides a cryptographic apparatus that saves computing operations or computation time while retaining or increasing the security, said apparatus comprising means for executing, with at least one exponentiation device, a computing step containing a modular exponentiation E E = xd (mod poq) with a first prime factor p, a second prime factor q, an exponent d and a base x, whereby b) means for performing a modular exponentiation two natural numbers r and s, and two numbers b1 and b2 in the interval [i,...,rl] and [1,,..s1] and relatively prime to r and s, respectively, are chosen, and whereby b1 and b2 fulfill the condition bt = b2 (mod ggT(r,s)), where ggT(r,s) designates the greatest common divisor of r and s, c) means for using the two numbers b1 and b2 to calculate according to the Chinese Remainder Theorem values X1 and x2 fulfilling the following conditions: x1 = x (modp) ,x1 = b1 (mod r) x2 = x (mod q), x2 = b2 (mod s) and then the following computing steps are performed: d_1 = d (mod p)) d_2 = d (mod (q)) z1 =x1 d1 (modpor) z2 = x2d1 (mod qos) and whereby .) represents the Eulerian function and kgV(r,s) the smallest d) means for calculating a number z from z1 and z2 according to the Chinese Re mainder Theorem with z=Z1 (mod por) ;z=z2 (mod qos); e) means for calculating the result E of the exponentiation by reduction of z modulo poq, f) means for checking the previously calculated number z (and thus automatically also the result E)for computing errors in a checking step, g) the checking step comprises the following computing operations: gl) calculating the numbers C1 = b1d1 (modor) C2=b2d2 (modos) whereby d_1 and d_2. are reduced before carrying out the modular exponentiation modulo r) and s), respectively, g2) comparing the values z1 and C1 modulo r as well as z2 and C2 modulo s, whereby the result of the modular exponentiation E is rejected as faulty if C1 ?Z1 mod r or C2 ? z2 mod s holds. 4E Dependent claims 3 to 12 and 15 to 24 show advantageous developments. As mentioned below, it is advantageous on certain arithmetic and logic units if a modulus in the modular exponentiation has many leading binary ones, so that different factors r and s signify a certain advantage here. Further, there are arithmetic and logic units optimized for modular exponentiation, but the mere data transfer from the central processing unit to the optimized arithmetic and logic unit for exponentiation causes considerable overhead. The present invention saves one exponentiation compared to the abovedescribed method with different factors r and s. According to the invention, two integers r and s are selected for example in the range of [0, 2kl] with 16 5 Now z1 = x d (modpof) and z2 = x d (mod qos) hold. According to the Chinese Remainder Theorem a'number z can easily be calculated from z1 and z2 with z = Z1 (mod por); z = z2 (mod qos) ;z = xd (mod pqkgV(r,s)) (17) The numbers r and j must according to the invention be chosen so that d is relatively prime to s)). Under these circumstances the Extended Euclidean Algorithm can be used to easily find a natural number e with 

Patent Number  200026  

Indian Patent Application Number  IN/PCT/2002/01333/KOL  
PG Journal Number  11/2007  
Publication Date  16Mar2007  
Grant Date  16Mar2007  
Date of Filing  25Oct2002  
Name of Patentee  GIESECKE & DEVRIENT GMBH,  
Applicant Address  PRINZREGENTENSTRASSE 159, 81677 MUNCHEN, ( A GERMANY COMPANY )  
Inventors:


PCT International Classification Number  G06F 7/72  
PCT International Application Number  PCT/EP01/05532  
PCT International Filing date  20010515  
PCT Conventions:
