A complete and explicit security reduction algorithm for RSA-based cryptosystems

In this paper, we introduce a conceptually very simple and demonstrative algorithm for finding small solutions (x,y) of ax + y = c mod N, where gcd(a, N) = 1. Our new algorithm is a variant of the Euclidian algorithm. Unlike former methods, it finds a small solution whenever such a solution exists. Further it runs in time O((log N)(3)), which is the same as the best known previous techniques, e.g. lattice-based solutions. We then apply our algorithm to RSA-OAEP and RSA-Paillier to obtain better security proofs. We believe that there will be many future applications of this algorithm in cryptography.