By David Kohel, Robert Rolland (ed.)
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Mathematical algorithms are crucial for all meeting language and embedded approach engineers who strengthen software program for microprocessors. This booklet describes thoughts for constructing mathematical exercises - from basic multibyte multiplication to discovering roots to a Taylor sequence. All resource code is on the market on disk in MS/PC-DOS layout.
Probably the most stated books in arithmetic, John Milnor's exposition of Morse thought has been crucial ebook at the topic for greater than 40 years. Morse idea used to be built within the Nineteen Twenties by way of mathematician Marston Morse. (Morse used to be at the college of the Institute for complex research, and Princeton released his Topological equipment within the idea of capabilities of a fancy Variable within the Annals of arithmetic reports sequence in 1947.
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Extra resources for Arithmetic, Geometry, Cryptography and Coding Theory 2009
Moreover, if v divides each of these two factors then in fact v | r. We are then led to consider two cases. First, suppose that v r. Then either v 2 | (t1 + t2 ) or v 2 | (r + t1 + t2 ). Let h = (g + 1)/2 . If m = deg v ≤ h, then by sieving we conclude that v 2 | pt for at most 2q g+1−2m+1 = 2q g+2−2m values of t. On the other hand, if m > h then deg v 2 > g + 1 so by sieving we now have v | pt for at most two values of t. Since the number of monic irreducible polynomials of degree m over k is bounded by q m /m, the number of values of t such that pt is 24 4 WOUTER CASTRYCK AND JOHN VOIGHT divisible by v 2 with v r is at most q g+1 q2 qh q h+1 (2q g+2−4 ) + · · · + (2q g+2−2h ) + 2 + ···+ 2 2 h h+1 g+1 g g+2−h h+1 g+1 q q q q + ···+ + + ··· + = 2 q g+1 + 2 h h+1 g+1 q(2q g+2−2 ) + = ≤ ≤ 2+ 2 g+1 h q g+1 + 2 i=2 q g+2−i + i q g+1 − 1 2 q g+1 + 2 2+ g+1 q−1 2 2 + q g+1 .
By repeating this technique, he can recover the secret scalar. This type of fault injection is also called computational safe-error attack. However, for the Montgomery ladder, the situation is diﬀerent as every intermediate result is used to compute the ﬁnal result. Hence, if the attacker induces a fault the ﬁnal result will inevitably be corrupted. Joye and Yen [JY02] proposed a slight modiﬁcation to the Montgomery ladder in order to make it resistant to M safe-error attacks, an attack that implies stronger assumptions in the attacker’s capabilities.
Each elliptic curve operation is implemented as the repetition of blocks of instructions that look alike in the power trace. The code of the scalar multiplication algorithm is then unrolled such that it appears as a repetition of the same atomic block. The sequence of blocks does not depend on the scalar used and their algorithm is then secure against SPA. A doubling in Jacobian coordinates is computed using 10 atomic blocks and 16 blocks for an addition, each atomic block costing 1M . However their construction uses dummy operations and can then be sensitive to fault attacks.