This paper describes an attack on the Rivest, Shamir and Adleman (RSA) cryptosystem utilizing the modulus $N=p^{2}q$ where p and q are two large balanced primes. Let $e_1,e_2<N^{\gamma }$ be the integers such that $d_1,d_2<N^{\delta }$ be their multiplicative inverses. Based on the two key equations $e_1{d}_1-k_1\varphi \left(N\right)=1$ and $e_2{d}_2-k_2\varphi \left(N\right)=1$ where $\varphi \left(N\right)=p(p-1)(q-1)$, our attack works when the primes share a known amount of least significant bits (LSBs) and the private exponents share an amount of most significant bits (MSBs). We apply the extended strategy of Jochemsz–May to find the small roots of an integer polynomial and show that N can be factored if $\delta <\frac{11}{10}+\frac{9}{4}\alpha -\frac{1}{2}\beta -\frac{1}{2}\gamma -\frac{1}{30}\sqrt{180\gamma +990\alpha -180\beta +64}.$ Our attack improves the bounds of some previously proposed attacks that makes the RSA variant vulnerable.

This paper presents a cryptanalytic approach on the variants of the RSA which utilizes the modulus N = p²q where p and q are balanced large primes. Suppose satisfying gcd(e, ϕ(N)) = 1 where ϕ(N) = p(p − 1)(q − 1) and d < N^δ be its multiplicative inverse. From ed − kϕ(N) = 1, by utilizing the extended strategy of Jochemsz and May, our attack works when the primes share a known amount of Least Significant Bits(LSBs). This is achievable since we obtain the small roots of our specially constructed integer polynomial which leads to the factorization of N. More specifically we show that N can be factored when the bound . Our attack enhances the bound of some former attacks upon N = p²q.

In this paper, we push further the cryptanalysis of a cryptosystem of the RSA’s variant which utilized a cubic Pell equation with the key equation ed − k(p2 + p + 1)(q2 + q + 1) = 1 where N = pq is an RSA modulus, e, N are publicized, while d, p, q are kept private. We consider the case where the prime factors share an amount of their least significant bits (LSBs), that is p and q satisfy p – q = 2m u where m is known, and u is unknown. Through this work, we show that via Coppermith’s method and lattice basis reduction, it is feasible to retrieve the secret key d and factor N for larger values of d.

The identification protocol is a type of zero-knowledge proof. One party (the prover) needs to prove his identity to another party (the verifier) without revealing the secret key to the verifier. One can apply the Fiat–Shamir transformation to convert an identification scheme into a signature scheme which can be used for achieving security purposes and cryptographic purposes, especially for authentication. In this paper, we recall an identification protocol, namely the RankID scheme, and show that the scheme is incorrect and insecure. Then, we proposed a more natural approach to construct the rank version of the AGS identification protocol and show that our construction overcomes the security flaws in the RankID scheme. Our proposal achieves better results when comparing the public key size, secret key size, and signature size with the existing identification schemes, such as Rank RVDC and Rank CVE schemes. Our proposal also achieves 90%, 50%, and 96% reduction for the signature size, secret key size, and public key size when compared to the Rank CVE signature scheme.