Related papers: Computing a Discrete Logarithm in O(n^3)

Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.

We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1) First we show that…

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…

We present an index calculus algorithm with double large prime variation which lends itself well to a rigorous analysis. Using this algorithm we prove that for fixed genus $g \geq 2$, the discrete logarithm problem in degree 0 class groups…

Bergman's Ring $E_p$, parameterized by a prime number $p$, is a ring with $p^5$ elements that cannot be embedded in a ring of matrices over any commutative ring. This ring was discovered in 1974. In 2011, Climent, Navarro and Tortosa…

In this paper we generalize the quantum algorithm for computing short discrete logarithms previously introduced by Eker{\aa} so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one…

Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are…

We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…

Let $p$ be a prime. In this paper, we compute complexities of some simple modules of symmetric groups labelled by two-part partitions. Most of the simple modules considered here are contained in the $p$-blocks with non-abelian defect…

We study the problem of multiway number partition optimization, which has a myriad of applications in the decision, learning and optimization literature. Even though the original multiway partitioning problem is NP-hard and requires…

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of…

This paper introduces the theory and hardware implementation of two new algorithms for computing a single component of the discrete Fourier transform. In terms of multiplicative complexity, both algorithms are more efficient, in general,…

In this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities. For these quadrature rules we derive error estimates in terms of $N$, the number of nodes, $k$ the…

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm…

We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the $G(n,p)$ random graph model, with number of nodes $n$ and…

Let $p$ be an odd prime and $R$ a $p$-torsion-free commutative $\mathbb{Z}_{(p)}$-algebra. We compute the periodic cyclic homology over $R$ of the universal differential graded algebra $R//p$ which is obtained from $R$ by universally…

Compressed manifold modes are locally supported analogues of eigenfunctions of the Laplace-Beltrami operator of a manifold. In this paper we describe an algorithm for the calculation of modes for discrete manifolds that, in experiments,…

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G cong H. For several decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the smallest prime…

This paper describes recent advances in the combinatorial method for computing $\pi(x)$, the number of primes $\leq x$. In particular, the memory usage has been reduced by a factor of $\log x$, and modifications for shared- and…

Approximating a definite integral of product of cosines to within an accuracy of n binary digits where the integrand depends on input integers x[k] given in binary radix, is equivalent to counting the number of equal-sum partitions of the…