Public key exponent attacks on multi-prime power modulus using continued fraction expansion method

Abstract

This paper proposes three public key exponent attacks of breaking the security of the prime power modulus 𝑁=𝑝2𝑞2 where 𝑝 and 𝑞 are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus 𝑁=𝑝2𝑞2 for q<𝑝<2q using key equation 𝑒𝑑−𝑘𝜙(𝑁)=1 where 𝜙(𝑁)= 𝑝2𝑞2(𝑝−1)(𝑞−1) can be broken by recovering the secret keys 𝑘 /𝑑 from the convergents of the continued fraction expansion of e/𝑁−2𝑁3/4 +𝑁1/2 . The paper also reports the second and third approaches of factoring 𝑛 multi-prime power moduli 𝑁𝑖=𝑝𝑖2 𝑞𝑖2 simultaneously through exploiting generalized system of equations 𝑒𝑖𝑑−𝑘𝑖𝜑(𝑁𝑖)=1 and 𝑒𝑖𝑑𝑖−𝑘𝜑(𝑁𝑖)=1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for 𝑖=1,2,…,𝑛.