Unfolded Modular Multiplication

Abstract

Sedlak’s [Sed] modular multiplication algorithm is one of the first real silicon implementations to speed up the RSA signature generation [RSA] on a smartcard, cf. [DQ]. Although it is nearly unknown in the scientific literature on cryptographic hardware it received in the practical smartcard world a considerable amount of interest, cf. [HP1, HP2, NMR]. The reason why it is so unknown might be given by the fact that the original publication was extremely hard to read and that Sedlak didn’t explain all the subtle implementation issues. Theoretically, Sedlak’s algorithm needs on average n/3 steps (i.e., additions/subtractions) to compute the modular product (α·βmod ν) for α, β and ν being n-bit numbers. The main result of this paper is that Sedlak’s algorithm can be practically speeded up by an arbitrary integral factor i ≥ 2, i.e., our new algorithm needs on average n/(3 · i) steps in order to compute the modular product (α·βmod ν). A further contribution of this paper is the mathematically proper and reader-friendly derivation of Sedlak’s algorithm leading naturally to our main result.