Related papers: Computing zeta functions of large polynomial syste…
Curves over finite fields are of great importance in cryptography and coding theory. Through studying their zeta-functions, we would be able to find out vital arithmetic and geometric information about them and their Jacobians, including…
We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the…
We investigate the analogues of certain classical estimates of Littlewood for the Riemann zeta-function in the context of quadratic Dirichlet $L$-functions over function fields. In some situations, we are actually able to establish finer…
We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upper-bounded by a polynomial in the worst case. We show that a natural generalization of Simon's…
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the…
This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…
In this paper,we develop a novel representation of the zeta function expressed as the limiting difference between two structured double sums. This approach leads to a new and elegant identity involving maximum functions and additive terms,…
We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions…
What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field…
We give a fast algorithm for computing the canonical basis of an irreducible highest-weight module for $U_q(\hat{\mathfrak{sl}}_e)$, generalising the LLT algorithm.
A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the…
Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.
A fast algorithm is presented for determining the linear complexity and the minimal polynomial of periodic sequences over GF(q) with period q n p m, where p is a prime, q is a prime and a primitive root modulo p2. The algorithm presented…
We present algorithms to perform modular polynomial multiplication or modular dot product efficiently in a single machine word. We pack polynomials into integers and perform several modular operations with machine integer or floating point…
Solving quadratic equations over finite fields is a fundamental task in algebraic coding theory and serves as a key subroutine for computing the roots of cubic and quartic polynomials. Notably, any quadratic polynomial over binary extension…
Polynomial functions on the group of units Q_n of the ring Z_{2^n} are considered. A finite set of reduced polynomials RP_n in Z[x] that induces the polynomial functions on Q_n is determined. Each polynomial function on Q_n is induced by a…
The Fast Fourier Transform (FFT) over a finite field $\mathbb{F}_q$ computes evaluations of a given polynomial of degree less than $n$ at a specifically chosen set of $n$ distinct evaluation points in $\mathbb{F}_q$. If $q$ or $q-1$ is a…
We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For…
We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.
Let $O$ be a one-dimensional Cohen-Macaulay local ring having a finite field as a coefficient field. The aim of this work is to extend the explicit computations of the St\"ohr Zeta Function of $O$ for one and two branches to an arbitrary…