38,648 research outputs found
Deterministic root finding over finite fields using Graeffe transforms
We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field F q . Our algorithms were designed to be particularly efficient in the case when the cardinality q − 1 of the multiplicative group of F q is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders
On Roots Factorization for PQC Algorithms
In this paper we consider several methods for an efficient extraction of roots of a polynomial over large finite fields. The problem of computing such roots is often the performance bottleneck for some multivariate quantum-immune cryptosystems, such as HFEv-based Quartz, Gui, etc. We also discuss a number of techniques for fast computation of traces as part of the factorization process. These optimization methods could significantly improve the performance of cryptosystems where roots factorization is a part thereof
On the Decoding Complexity of Cyclic Codes Up to the BCH Bound
The standard algebraic decoding algorithm of cyclic codes up to the
BCH bound is very efficient and practical for relatively small while it
becomes unpractical for large as its computational complexity is .
Aim of this paper is to show how to make this algebraic decoding
computationally more efficient: in the case of binary codes, for example, the
complexity of the syndrome computation drops from to , and
that of the error location from to at most .Comment: accepted for publication in Proceedings ISIT 2011. IEEE copyrigh
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
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