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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 complexity and $2$-adic complexity of $(s_n)$. We show that for a typical $p$ the sequence $(s_n)$ is quite unbalanced. However, there are still infinitely many $p$ such that $(s_n)$ is very balanced. We also prove similar results for the distribution of longer patterns. Moreover, we give general lower bounds on the linear complexity and $2$-adic complexity of~$(s_n)$ and state sufficient conditions for attaining their maximums. Hence, for carefully chosen $p$, these sequences are attractive candidates for cryptographic applications.
For a prime $pge 5$ let $q_0,q_1,ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $
Fermat-Euler quotients arose from the study of the first case of Fermats Last Theorem, and have numerous applications in number theory. Recently they were studied from the cryptographic aspects by constructing many pseudorandom binary sequences, whos
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
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
Let tau(.) be the Ramanujan tau-function, and let k be a positive integer such that tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set {1,...,k}. Th