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Schindler s Attack
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Schindler [lag] gives a timing attack that succeeds against RSA implementations that employ repeated squaring and both CRT and Montgomery niultiplication (but not both Karatsuba multiplication and long multiplication). First, we describe the precise modular exponentiation scenario for which Schindler s attack will succeed. Then we discuss Schindler s attack in some det ai1. We assume that the Montgomery multiplication algorithm is implemented as given in Table 7.6. The repeated squaring algorithm using Montgomery multiplication is given in Table 7.7. Table 7.6: Montgomcry Multiplication
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Find Montgomery product n b , where a = aR (mod N ) and b = bR (mod N ) Given RR - NN = 1 Montgomery(a , 6 ) z = a b T = ( 2 (mod R ) ) N (mod R) s = ( z + r N ) / R (mod N ) i f s 2 N then s = s - AT // extra reduction
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Table 7.7: Repeated Squaring with Montgomery Multiplication
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Suppose that the RSA system we want to attack uses the repeated squaring algorithm in Table 7.7 (which relies on the Montgomery multiplication algorithm in Table 7.6). Also, suppose that the RSA system uses CRT. Then for each mod N reduction, where N = p q , we compute a mod p reduction and a mod q reduction, using the algorithm in Table 7.7 for both. We combine these two results to obtain the desired mod N reduction, as discussed above in Section 7.4.1. We assume that the attacker is able to choose ciphertext messages Cj and accurately time the decryption, that is, the computation C j (mod N ) . Of course, the objective is to determine the private key d. Schindler s timing attack [lag] takes advantage of the extra reduction step in the Montgomery algorithm. Schindler derives precise probabilities that an extra reduction occurs when using the Montgomery algorithm. Suppose that we compute Montgomery(a , B ) using the algorithm in Table 7.6, assuming that a = aR (mod N ) and B is randomly-selected in {0,1,2,. . . , N - 1). Then Schindler shows that for each application of the Montgomery algorithm, the probability of an extra reduction is
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This gives us a useful probability for an extra reduction in the multiply step of the repeated squaring algorithm in Table 7.7. For the square step, where the element to be squared, say B , is selected at random in {0,1,2,. . . ,N - l}, Schindler is able to show that P(extra reduction in Montgomery(B, B ) )= -. 3R
When computing a modular exponentiation ud (mod N ) using the CRT approach, we first compute ad, (mod p ) using the repeated squaring algorithm in Table (7.7), where d, = d (mod ( p - 1)). Suppose that when computing u d p (mod p ) , we have ko multiply steps and kl squaring steps. Note that ko and kl depend only on d, and, therefore, only on d and p , and not on a. Since the probability (7.21) holds for each multiply, and the probability (7.22) holds for each square, the expected number of extra reductions is a (mod p ) P (7.23) +k1-. IC0 2R 3R As a function of a , the expression in (7.23) is piecewise linear-more precisely, it is a linear function with discontinuities a t integer multiples of p . Qualitatively, the graph of (7.23) is similar to that in Figure 7.2 (see Problem 2). Note that the total number of extra reductions in the calculation of Cd (mod N) also include extra reductions due to the factor q. Nevertheless, there would still be a discontinuity in the total number of extra reductions at every integer multiple of p (and also q ) .