Conference Agenda

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Session Overview
Session
MS132, part 3: Polynomial equations in coding theory and cryptography
Time:
Wednesday, 10/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-123
52 seats, 100m^2

Presentations
3:00pm - 5:00pm

Polynomial equations in coding theory and cryptography

Chair(s): Alessio Caminata (University of Neuchâtel, Switzerland), Alberto Ravagnani (University College Dublin, Ireland)

Polynomial equations are central in algebraic geometry, being algebraic varieties geometric manifestations of solutions of systems of polynomial equations. Actually, modern algebraic geometry is based on the use of techniques for studying and solving geometrical problems about these sets of zeros. At the same time, polynomial equations have found interesting applications in coding theory and cryptography. The interplay between algebraic geometry and coding theory is old and goes back to the first examples of algebraic codes defined with polynomials and codes coming from algebraic curves. More recently, polynomial equations have found important applications in cryptography as well. For example, in multivariate cryptography, one of the prominent candidates for post-quantum cryptosystems, the trapdoor one-way function takes the form of a multivariate quadratic polynomial map over a finite field. Furthermore, the efficiency of the index calculus attack to break an elliptic curve cryptosystem relies on the effectiveness of solving a system of multivariate polynomial equations. This session will feature recent progress in these and other applications of polynomial equations to coding theory and cryptography.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Classical and Quantum Evaluation Codes at the Trace Roots

Diego Ruano
University of Valladolid

We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field with four elements which are records at http://codetables.de/. Joint work with C. Galindo and F. Hernando (Jaume I University).

 

Optimal curves and codes with locality

Gretchen Matthews
Virginia Tech

In some applications, it is desirable to have erasure codes that have recovery algorithms for a relatively large number of missing pieces (erasures). To maintain data availability at all times, it is advantageous to recover information at one node, which may fail or be offline, by accessing a small number of other nodes. This leads to the notion of local recovery, meaning that for a code C of length n, a codeword symbol can be recovered by accessing at most r other coordinates of the codeword; the code C is then said to have locality r. Though there are tradeoffs in terms of the rate and minimum distance, one typically wants r small, so that communications of information from other locations is minimal, hence saving communications bandwidths. In addition, it is often desirable for each coordinate to have multiple recovery sets; such a code is said to have availability. In this talk, we consider codes with locality and availabilty constructed from optimal curves.

 

The Story of Solving Random Quadratic Multivariate Systems of Equations

Bo Yin Yang
Academia Sinica

Solving quadratic multivariate systems over finite fields is one of the fundamental problem in computer science and cryptography. In fact, Shannon is said to have remarked that breaking a good cipher should be as hard as solving a system of nonlinear equations. Exactly how hard that really is has been an interesting open problem. We discuss the interesting history and recent developments in solving multivariate quadratic systems, particularly that over GF(2).

 

The Zeta Function for Generalized Rank Weights

Eimear Byrne, Giuseppe Cotardo, Alberto Ravagnani
University College Dublin

The zeta function of a linear block code with the Hamming metric encodes its weight distribution in a convenient way. It is particularly useful to analyze the structural properties of a family of codes that share the same weight enumerator. The definition of the zeta function is motivated by the properties of codes with the Hamming weight obtained from algebraic curves via Goppa's construction. The rank-metric analogue of the zeta function is defined as the generating function of the normalized q-binomial moments of a matrix code endowed with the rank distance. This algebraic object is a code invariant with respect to puncturing and shortening operations, and links the rank distribution of codes to a Riemann-type hypothesis in the context of coding theory.

In the first part of the talk we present the main definitions and results on the theory of rank-metric zeta functions. We then extend this concept to generalized distributions of matrix codes, and discuss the duality theory of these. In particular, we present a generalized version of the MacWilliams identities for rank-metric codes, and prove some rigidity properties of extremal codes with respect to generalized distributions.

(the new results in this talk are joint work with E. Byrne and A. Ravagnani)