Cryptanalysis of RSA variants with modified euler quotient

Abstract

The standard RSA scheme provides the key equation ed≡1(modφ(N)) for N= pq, where φ(N) = (p- 1) (q- 1) is Euler quotient (or Euler’s totient function), e and d are the public and private keys, respectively. It has been extended to the following variants with modified Euler quotient ω(N) = (p2- 1) (q2- 1), which in turn indicates the modified key equation is ed≡1(modω(N)). An RSA-type scheme based on singular cubic curves y2≡x3+bx2(modN) for N= pq.An extended RSA scheme based on the field of Gaussian integers for N= PQ, where P, Q are Gaussian primes with p= | P|, q= | Q|.A scheme working in quadratic field quotients using Lucas sequences with an RSA modulus N= pq. In this paper, we investigate some key-related attacks on such RSA variants using lattice-based techniques. To be specific, small private key attack, multiple private keys attack, and partial key exposure attack are proposed. Furthermore, we provide the first results for multiple private keys attack and partial key exposure attack when analyzing the RSA variants with modified Euler quotient.