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In a paper of P. Paillier and J. Villar a conjecture is made about the malleability of an RSA modulus. In this paper we present an explicit algorithm refuting the conjecture. Concretely we can factorize an RSA modulus n using very little information on the factorization of a concrete n coprime to n. However, we believe the conjecture might be true, when imposing some extra conditions on the auxiliary n allowed to be used. In particular, the paper shows how subtle the notion of malleability is.
Fix $a in mathbb{Z}$, $a otin {0,pm 1}$. A simple argument shows that for each $epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{frac12-epsilon}$. It is an open problem to show t
Let $1<g_1<ldots<g_{varphi(p-1)}<p-1$ be the ordered primitive roots modulo~$p$. We study the pseudorandomness of the binary sequence $(s_n)$ defined by $s_nequiv g_{n+1}+g_{n+2}bmod 2$, $n=0,1,ldots$. In particular, we study the balance, linear comp
We prove that $|x-y|ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the primitive element problem for two singular moduli. In a previous article Faye and Riffaut show that the
Let $p=2n+1$ be an odd prime, and let $zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $overline{mathbb{Q}_p}$ of $mathbb{Q}_p$. We let $ginmathbb{Z}_p[zeta_{p^2-1}]$ be a primitive root modulo $pmathbb{Z}_p[zeta_{p^2
Let $qgeq 1$ be any integer and let $ epsilon in [frac{1}{11}, frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ pequiv 1pmod{q}, quad loglog p > frac{log 6.83}{frac{1}{2}-epsilon} mbox{ and } fra