New attacks on prime power N = prq using good approximation of φ(N)

• Main Authors: Shehu, Sadiq, Kamel Ariffin, Muhammad Rezal
• Format: Article
• Language: English
• Published: Institute for Mathematical Research, Universiti Putra Malaysia 2017
• Online Access: http://psasir.upm.edu.my/id/eprint/58180/1/121-138.pdf; http://psasir.upm.edu.my/id/eprint/58180/; http://einspem.upm.edu.my/journal/fullpaper/vol11saugust/121-138.pdf

This paper proposes three new attacks. Our first attack is based on the RSA key equation ed − kφ(N) = 1 where φ(N) = pr-1(p-1)(q-1). Let q <p <2q and 2p 3r+2/r+1 |p r-1/r+1 – q r-1/r+1| < 1/6Ny with d = Nδ. If δ < 1-y/2 we shows that k/d can be recovered among the convergents of the continued fractions expansions of e/N-2N r/r+1 + N r-1/r+1. We furthered our analysis on j prime power moduli Ni = priqi satisfying a variant of the above mentioned condition. We utilized the LLL algorithm on j prime power public keys (Ni, ei) with Ni = priqi and we were able to factorize the j prime power moduli Ni = priqi simultaneously in polynomial time.