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(abR)R' = 6 . 64 = 384 = 68 (mod 79).
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ab (mod N )
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6 1 . 5 = 305 = 68 (mod 79).
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Montgomery multiplication is certainly more work than it is worth in the simple example considered above. However, suppose that instead of ~0x1puting ab (mod N ) , we want to compute ad (mod N). Then to use the
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Montgomery algorithm, we must pay the price of converting a into Montgomery form, but having done so, all of the multiplications required in the computation of ad (mod N ) can be computed using (7.18) and (7.19) (and extra reductions, as required), without any expensive division operations. The final result must be converted from Montgomery form back into nonMontgomery form, which requires one additional mod N operation. The bottom line is that only two expensive mod N operations are required, since the multiplications are all computed in Montgomery form which only requires efficient mod R operations. With respect to the timing attacks discussed below, the extra reduction step provides a crucial timing difference that an attacker can exploit in some circumstances. Other tricks are also used to speed up modular exponentiation. Of these, the sliding window and Karatsuba multiplication are the most significant. A sliding window is a straightforward time-memory trade-off applied to the repeated squaring algorithm. That is, instead of processing each bit individually, we process the bits in blocks (say, blocks of five consecutive bits) and use pre-computed tables containing the required factors. Karatsuba multiplication [76] is the most efficient method to multiply two numbers with the same number of digits-assuming that addition is much cheaper than multiplication. The work factor for Karatsuba multiplication is multiplications, where n is the number of bits on the order of nlog2 = n1.585 in each of the numbers to be multiplied, whereas normal long multiplication has a work factor on the order of n2. The Karatsuba algorithm is based on a simple observation. The naive approach to computing the product (a0 a1 . 10)(b, bl . 10) is
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+ + (a0+ a1 . lo)@, + bl . 10) = aobo + (a& + U l b 0 ) l O + albl .102,
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which requires four multiplications to determine the coefficients of the powers of ten. However, the same can be accomplished with just three multiplications. since
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+ a1 . 10)(bo + bl . 10)
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+ [(a0+ a l ) ( b o + b l )
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~ o b o ~ l b l ] l O albl . lo2 (7.20) -
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and this is the essential idea behind Karatsuba multiplication. The Karatsuba technique can be used for numbers of any magnitude. For example, suppose that we want to find the product
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( c ~ c1 . l o
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+ c2 . lo2 + c : .~103)(do+ dl . l o + d2 . lo2 + d 3 . lo3).
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We can rewrite the first term a s
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+ c1 . 10 and C1 = c2 + c3 . 10. Similarly, we can rewrite the
product is given by
+ dl 10) + (d2 + d3.10)102 = Do + D1 . l 0 2 , where Do = do + d l . 10 and D1 = d2 + d3 . 10. In this case, the Karatsuba
+c 1
+ = C o ~+ [(coC ~ ) ( + a ) cono c 1 ~ l ] 1 0+ c l ~l o 4 . o + D ~ 2 . ,
102)(Do D1 102)
Here, the three products involving the Ci and 0 3 are computed as in (7.20). Consequently, given any product, we can recursively apply the Karatsuba multiplication technique. At each step in the recursion, three multiplications are required, and the numbers are half as big as at the previous step. A straightforward analysis yields the claimed work factor of n1.585. Note that the Karatsuba algorithm holds if the base 10 (or l o 2 ) is replaced by any other base. Also, the algorithm is most efficient if the two numbers to be multiplied are of about the same magnitude. At this point, we have more than enough background to discuss the three timing attacks mentioned above. First, we consider Kocher s attack, which only applies to systems that use repeated squaring, but not CRT or Montgomery multiplication. Kocher s attack has been successfully applied to smartcards. Then we discuss Sdiindler s method, which can be used when CRT and Montgomery multiplication are employed. Finally, we present the justifiably famous Brumley-Boneh attack, which succeeds against RSA as implemented in a version of OpenSSL in a realistic scenario (over a network). The OpenSSL implementation of RSA is highly optimized, using CRT, Montgomery multiplication, sliding windows and Karatsuba s algorithm. As of this writing, the Brurnley-Boneh attack stands as the greatest success in the relatively young field of timing attacks. We note in passing that timing attacks have recently been directed at symmetric ciphers [la] but, to date, these have proven far less of a realistic threat than timing attacks on public kcy cryptosystenis.