Univariate polynomial factorization over finite fields

We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results: (i) we decrease from O(n 2 + n 1+o(1) log q) to O(n 1.9998 + n 1+o(1) log q) the known arithmetic complexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n × n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii) we decrease from O(m 1.575 n) to O(m 1.5356 n) the known bound for computing basic solutions to a linear programming problem with m constraints and n variables. Keywords: rectangular matrix multiplication, asymptotic arithmetic complexity, bilinear algorithm, polynomial factorization over finite fields MSC(2000): 68Q25, 11Y16

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“…To prove (28), first note that out of the G compatible block products, the number of unordered block product pairs (…”

Section: Lemma 3 Let Us Give a Small ε A Large Odd M And A Salem-spmentioning

confidence: 99%

Fast rectangular matrix multiplication and some applications

Ben-sheng

Han

et al. 2008

Sci. China Ser. A-Math.

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“…This algorithm is probabilistic in nature. A detailed description of both methods can be found in [13].…”

Section: Irreducible Polynomialsmentioning

confidence: 99%

Using irreducible polynomials for random number generation

Herendi¹,

Major²

2023

AMI

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A method is presented for generating random numbers with uniform distribution using linear recurrence sequences with very large period lengths. This method requires an irreducible polynomial modulo 2 to define the sequence. A suitable method for generating an infinite number of such polynomials is presented. The polynomials generated in this way can have an arbitrarily large degree, and a large enough order to make them suitable for practical applications.

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“…The first step represents at most log(p) squares and multiplications, and the second step represents at most n iterations of the Euclidean algorithm. The roots are found by factorising the polynomial gcd(F (X), E(X)) mod p. Some examples of factorization algorithms can be found in [19]. Proof.…”

Section: Number Of Pmns In the General Casementioning

confidence: 99%

On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$

Bajard,

Marrez,

Plantard

et al. 2020

Preprint

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Polynomial Modular Number System (PMNS) is a convenient number system for modular arithmetic, introduced in 2004. The main motivation was to accelerate arithmetic modulo an integer p. An existence theorem of PMNS with specific properties was given. The construction of such systems relies on sparse polynomials whose roots modulo p can be chosen as radices of this kind of positional representation. However, the choice of those polynomials and the research of their roots are not trivial.In this paper, we introduce a general theorem on the existence of PMNS and we provide bounds on the size of the digits used to represent an integer modulo p.Then, we present classes of suitable polynomials to obtain systems with an efficient arithmetic. Finally, given a prime p, we evaluate the number of roots of polynomials modulo p in order to give a number of PMNS bases we can reach. Hence, for a fixed prime p, it is possible to get numerous PMNS, which can be used efficiently for different applications based on large prime finite fields, such as those we find in cryptography, like RSA, Diffie-Hellmann key exchange and ECC (Elliptic Curve Cryptography).

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Univariate polynomial factorization over finite fields

Cited by 7 publications

References 94 publications

Fast rectangular matrix multiplication and some applications

Fast rectangular matrix multiplication and some applications

Using irreducible polynomials for random number generation

On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$

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