On the discrete logarithm problem in finite fields of fixed characteristic

For $q$ a prime power, the discrete logarithm problem (DLP) in $\mathbb{F}_{q}$ consists in finding, for any $g \in \mathbb{F}_{q}^{\times}$ and $h \in \langle g \rangle$, an integer $x$ such that $g^x = h$. We present an algorithm for computing discrete logarithms with which we prove that for each prime $p$ there exist infinitely many explicit extension fields $\mathbb{F}_{p^n}$ in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions $\mathbb{F}_{p^n}$ in expected quasi-polynomial time.Comment: 15 pages, 2 figures. To appear in Transactions of the AM

Rational Transformations and Invariant Polynomials

Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with (normalized) generators of the field $K(x)^G$ of $G$-invariant rational functions for $G$ a finite subgroup of $\operatorname{PGL}_2(K)$, where $K$ is an arbitrary field. Our main theorem shows that the factorization is related to a well-known group action of $G$ on a subset of monic polynomials. With this, we are able to extend a result by Lucas Reis for $G$-invariant irreducible polynomials. Additionally, some new results about the number of irreducible factors of rational transformations for $Q$ a generator of $\mathbb{F}_q(x)^G$ are given when $G$ is non-cyclic

Explicit Subcodes of Reed-Solomon Codes that Efficiently Achieve List Decoding Capacity

In this paper, we introduce a novel explicit family of subcodes of Reed-Solomon (RS) codes that efficiently achieve list decoding capacity with a constant output list size. Our approach builds upon the idea of large linear subcodes of RS codes evaluated on a subfield, similar to the method employed by Guruswami and Xing (STOC 2013). However, our approach diverges by leveraging the idea of {\it permuted product codes}, thereby simplifying the construction by avoiding the need of {\it subspace designs}. Specifically, the codes are constructed by initially forming the tensor product of two RS codes with carefully selected evaluation sets, followed by specific cyclic shifts to the codeword rows. This process results in each codeword column being treated as an individual coordinate, reminiscent of prior capacity-achieving codes, such as folded RS codes and univariate multiplicity codes. This construction is easily shown to be a subcode of an interleaved RS code, equivalently, an RS code evaluated on a subfield. Alternatively, the codes can be constructed by the evaluation of bivariate polynomials over orbits generated by \emph{two} affine transformations with coprime orders, extending the earlier use of a single affine transformation in folded RS codes and the recent affine folded RS codes introduced by Bhandari {\it et al.} (IEEE T-IT, Feb.~2024). While our codes require large, yet constant characteristic, the two affine transformations facilitate achieving code length equal to the field size, without the restriction of the field being prime, contrasting with univariate multiplicity codes.Comment: 20 page

The Goldman-Rota identity and the Grassmann scheme

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The main step is a constructive, linear algebraic interpretation of the Goldman-Rota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using this we inductively construct an explicit orthogonal symmetric Jordan basis with respect to the up operator and write down the singular values, i.e., the ratio of the lengths of the successive vectors in the Jordan chains. The collection of all vectors in this basis of a fixed rank forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.Comment: 19 Page

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation