Increment of insecure RSA private exponent bound through perfect square RSA diophantine parameters cryptanalysis

• Published in: Computer standards and interfaces Vol. 80; p. 103584
• Main Authors: Wan Mohd Ruzai, Wan Nur Aqlili, Nitaj, Abderrahmane, Kamel Ariffin, Muhammad Rezal, Mahad, Zahari, Asbullah, Muhammad Asyraf
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2022; Elsevier BV; Elsevier
• Subjects: Algebraic cryptanalysis; Algorithms; Computer Science; Cryptography; Cryptography and Security; Diophantine approximation; Diophantine equation; Integer factorization problem; Kleptography; Lattice basis reduction; Parameters; Polynomials; RSA cryptosystem; Diophantine approximation; Algebraic cryptanalysis; Integer factorization problem; Lattice basis reduction; Kleptography; RSA cryptosystem; lattice basis reduction; kleptography; algebraic cryptanalysis; integer factorization problem
• ISSN: 0920-5489; 1872-7018; 1872-7018
• DOI: 10.1016/j.csi.2021.103584

•A new weak RSA key equation structure that solves the factoring problem under certain specified conditions in polynomial time is proposed.•Note that our cryptanalytic work extends the bound of insecure RSA decryption exponents of some previous literature.•Experimental results are provided to demonstrate the effectiveness of the new attack. The public parameters of the RSA cryptosystem are represented by the pair of integers N and e. In this work, first we show that if e satisfies the Diophantine equation of the form ex2−ϕ(N)y2=z for appropriate values of x,y and z under certain specified conditions, then one is able to factor N. That is, the unknown yx can be found amongst the convergents of eN via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given (Ni,ei) for i=1,2,⋯,k and a fixed x that fulfills the Diophantine equation eix2−yi2ϕ(Ni)=zi. This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.