IA Scholar Query: An approach to an arithmetic analog of Berlekamp's algorithm.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSun, 27 Feb 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On Efficient Noncommutative Polynomial Factorization via Higman Linearization
https://scholar.archive.org/work/7cxxi5fxvzhlzi4j6p3a5x46pe
In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F of polynomials in noncommuting variables x_1,x_2,..., x_n over the field F. We obtain the following result: Given a noncommutative arithmetic formula of size s computing a noncommutative polynomial f in F as input, where F=F_q is a finite field, we give a randomized algorithm that runs in time polynomial in s, n and log q that computes a factorization of f as a product f=f_1f_2\cdots f_r, where each f_i is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f. We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.V. Arvind, Pushkar S. Joglekarwork_7cxxi5fxvzhlzi4j6p3a5x46peSun, 27 Feb 2022 00:00:00 GMTUnderstanding binary-Goppa decoding
https://scholar.archive.org/work/vi6w4bw4bvcgjobyatfxh5uut4
This paper reviews, from bottom to top, a polynomial-time algorithm to correct t errors in classical binary Goppa codes defined by squarefree degree-t polynomials. The proof is factored through a proof of a simple Reed-Solomon decoder, and the algorithm is simpler than Patterson's algorithm. All algorithm layers are expressed as Sage scripts. The paper also covers the use of decoding inside the Classic McEliece cryptosystem, including reliable recognition of valid inputs.Daniel J. Bernsteinwork_vi6w4bw4bvcgjobyatfxh5uut4