while the corresponding variance is var(T(Cj) - i 0 . . . 3 ) = (12 O2 in .NET

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Although thc mean is the same in both cases, Kocher s attack tells us that the smaller variance indicates that dodld2d3 = 1010 is the correct answer. But this begs the question of why we should observe a smaller variance in case of a correct guess for dodld td3.
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Consider T(Cj), the timing of a particular computation Cf (mod N ) in Table 7.5. As above, for this T(C,y), let be the emulated timing for the square and multiply steps corresponding to the l t h bit of the private exponent . Also, let tp be the actual timing of the square and multiply steps corresponding to the t t h bit of the private exponent. Let u include all timing not accounted for in the te. The value u can be viewed as representing the measurement error . In the example above, we assumed the private exponent d is eight bits, so for this case
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T ( C j )= t o
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Now suppose that the high-order bits of d are dodld2d3 = 1010. Then for the timing T ( C j )we have var(T(Cj) - to...:<) var(t4) =
7.4 RSA IMPLEMENTATIONATTACKS
since = te, for e = 0 , 1 , 2 , 3 and, consequently, there is no variance due to these emulated timings it. Note that here, we are assuming the te are independent and that the measurement error u is independent of the te, which appear to be valid assumptions. If we denote the common variance of each te by var(t), we have var(T(Cj) - to...3)
= 4var(t)
+ var(u)
However, if dodld2d3 = 1010, but we emulate dodld2d3 = 1001, then from the point of the first d j that is in error, our emulation will fail, giving us essentially random timing results. In this case, the first emulation error occurs at d2 so that we find var(T - to...3) = var(t2
6var(t)
+ var(u),
+ var(t3 - t 3 ) + var(t4) + var(t5) + var(t6) + var(t7) + var(u)
since the emulated timings & and i3 can vary from the actual timings t 2 and t 3 , respectively. Although conceptually simple, Kocher s technique gives a powerful and practical approach to conducting a timing attack on an RSA implementation that uses repeated squaring (but not CRT or Montgomery multiplication). For the attack to succeed, the variance of the error term u must not vary too greatly between the different cases that are tested. Assuming that a simple repeated squaring algorithm is employed, this would almost certainly be the case since u only includes loop overhead and timing error. For more advanced modular exponentiation techniques, var (u) could differ greatly for different emulated bits, effectively masking the timing information needed to recover the bits of d. The amount of data required for Kocher s attack (that is, the number of chosen decryptions that must be timed) depends on the error term u. Note that timings can be reused as bits of d are determined, since, given additional bits of d , only the emulation steps need t o change. Therefore, the required number of timings is not nearly as daunting as it might appear at first blush. The major limitation to Kocher s attack is that repeated squaring, without CRT or Montgomery multiplication, is only used in RSA implementations in highly resource-constrained environments, such as smartcards. In [85], Kocher argues that his timing attack-as discussed in this section--should work for RSA implementations that employ CRT. However, Schindler [129] (among others) disputes this assertion. The next two timing attacks we discuss will succeed against RSA implementations that utilize more highly optimized modular exponentiation techniques.