By David Kohel, Robert Rolland (ed.)

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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.