Factoring the Modulus of Type <i>N</i> = <i>p</i>²<i>q</i> by Finding Small Solutions of the Equation <i>e</i><i>r</i> − (<i>N</i><i>s</i> + <i>t</i>) = <i>α</i><i>p</i>² + <i>β</i><i>q</i>²

Abstract

The modulus of type N=p2q is often used in many variants of factoring-based cryptosystems due to its ability to fasten the decryption process. Faster decryption is suitable for securing small devices in the Internet of Things (IoT) environment or securing fast-forwarding encryption services used in mobile applications. Taking this into account, the security analysis of such modulus is indeed paramount. This paper presents two cryptanalyses that use new enabling conditions to factor the modulus N=p2q of the factoring-based cryptosystem. The first cryptanalysis considers a single user with a public key pair (e,N) related via an arbitrary relation to equation er−(Ns+t)=αp2+βq2, where r,s,t are unknown parameters. The second cryptanalysis considers two distinct cases in the situation of k-users (i.e., multiple users) for k≥2, given the instances of (Ni,ei) where i=1,…,k. By using the lattice basis reduction algorithm for solving simultaneous Diophantine approximation, the k-instances of (Ni,ei) can be successfully factored in polynomial time.

Keywords

• cryptography
• IoT security
• lattice basis reduction
• Diophantine approximation
• pre-quantum cryptography