Related papers: Efficient Lifting of Discrete Logarithms Modulo Pr…

We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial.…

We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that…

We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was…

E. Bach, following an idea of T. Itoh, has shown how to build a small set of numbers modulo a prime p such that at least one element of this set is a generator of $\pF{p}$\cite{Bach:1997:sppr,Itoh:2001:PPR}. E. Bach suggests also that at…

For fixed positive integers $n$, we study the solution of the equation $n = k + p_k$, where $p_k$ denotes the $k$th prime number, by means of the iterative method \[ k_{j+1} = \pi(n-k_j), \qquad k_0 = \pi(n), \] which converges to the…

We propose various strategies for improving the computation of discrete logarithms in non-prime fields of medium to large characteristic using the Number Field Sieve. This includes new methods for selecting the polynomials; the use of…

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Ap\'ery numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there…

Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n. For n large compared to p, we establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the iterated…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

Designing a deterministic polynomial time algorithm for factoring univariate polynomials over finite fields remains a notorious open problem. In this paper, we present an unconditional deterministic algorithm that takes as input an…

This is an exposition, for pedagogical purposes, of the formal power series proof of Bostan, Christol and Dumas [3] of the result stated in the title (a corollary of the Christol theorem).

We present SGDPLL(T), an algorithm that solves (among many other problems) probabilistic inference modulo theories, that is, inference problems over probabilistic models defined via a logic theory provided as a parameter (currently,…

We introduce a new deterministic factoring algorithm, which could be described in the cryptographically fashionable term of "factoring with hints": we show that, given the knowledge of the factorisations of $O(N^{1/3+\epsilon})$ terms…

In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical…

We prove that all endo-$p$-permutation modules for a finite group are liftable from characteristic $p>0$ to characteristic $0$.

It is well-known that the congruence $\sum_{i=1}^{ n} i^{ n} \equiv 1 \pmod{n}$ has exactly five solutions: $\{1,2,6,42,1806\}$. In this work, we characterize the solutions to the congruence $1^n + 2^n + \dotsb + n^n\equiv p \pmod{n}$ for…

We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a…

We lift the $5$-dimensional characteristic $3$ representation of $M_{11}$ to a complex representation of the amalgam ${\rm GL}(2,3)*_{D_8}S_{4}$, and consider its reduction (mod $p$) for other odd primes.

The classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method…

Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the…