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تسجيل مستخدم جديدFinite sets containing near-primitive roots
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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 the same result with $frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{frac{1}{2}+frac{1}{30}}$. The same method allows one to produce, for each $epsilon > 0$, explicit finite sets $mathcal{A}$ with the property that for almost all primes $p$, some element of $mathcal{A}$ has order exceeding $p^{1-epsilon}$. Similar results hold for orders modulo general integers $n$ rather than primes $p$.
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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.
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
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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
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By definition primitive and $2$-primitive elements of a finite field extension $mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primiti
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