Discrete logarithm computations over finite fields using Reed-Solomon codes

Cheng and Wan have related the decoding of Reed-Solomon codes to the computation of discrete logarithms over finite fields, with the aim of proving the hardness of their decoding. In this work, we experiment with solving the discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2) operations over GF(q), operating on a q x q matrix with (h+2) q non-zero coefficients. We give faster variants including an incremental version and another one that uses auxiliary finite fields that need not be subfields of GF(q^h); this variant is very practical for moderate values of q and h. We include some numerical results of our first implementations

Discrete Logarithm Factory

The Number Field Sieve and its variants are the best algorithms to solve the discrete logarithm problem in finite fields. The Factory variant accelerates the computation when several prime fields are targeted. This article adapts the Factory variant to non-prime finite fields of medium and large characteristic. We combine this idea with two other variants of NFS, namely the tower and special variant. This combination leads to improvements in the asymptotic complexity. Besides, we lay out estimates of the practicality of this method for 1024-bit targets and extension degree $6$

Algebraic and analytic techniques in coding theory

Error correcting codes are designed to tackle the problem of reliable trans- mission of data through noisy channels. A major challenge in coding theory is to efficiently recover the original message even when many symbols of the received data have been corrupted. This is called the unique decoding problem of error correcting codes. More precisely, if the user wants to send K bits, the code stretches K bits to N bits to tolerate errors in the N bits. Then the goal is to recover the original K bits of the message. Often, the receiver requires only a certain part of the message. In such cases, analyzing the entire received data (word) becomes prohibitive. The challenge is to design a local decoder which queries only few locations of the received word and outputs the part of the message required. This is known as local decoding of an error correcting code. The unique decoding problem faces a certain combinatorial barrier. That is, there is a limit to the number of errors it can tolerate in order to uniquely identify the correct message. This is called the unique decoding radius. A major open problem is to understand what happens if one allows for errors beyond this threshold. The goal is to design an algorithm that can recover the right message, or possibly a list of messages (preferably a small number). This is referred to as list decoding of an error correcting code. At the core of many such codes lies polynomials. Polynomials play a fundamental role in computer science with important applications in algorithm design, complexity theory, pseudo-randomness and machine learning. In this dissertation, we improve our understanding of well known classes of codes and discover various properties of polynomials. As an additional consequence, we obtain results in a suite of problems in effective algebraic geometry, including Hilbert’s nullstellensatz, ideal membership problem and counting rational points in a variety.Computer Science

Statistical methods for sparse functional object data: elastic curves, shapes and densities

Many applications naturally yield data that can be viewed as elements in non-linear spaces. Consequently, there is a need for non-standard statistical methods capable of handling such data. The work presented here deals with the analysis of data in complex spaces derived from functional L2-spaces as quotient spaces (or subsets of such spaces). These data types include elastic curves represented as d-dimensional functions modulo re-parametrization, planar shapes represented as 2-dimensional functions modulo rotation, scaling and translation, and elastic planar shapes combining all of these invariances. Moreover, also probability densities can be thought of as non-negative functions modulo scaling. Since these functional object data spaces lack a natural Hilbert space structure, this work proposes specialized methods that integrate techniques from functional data analysis with those for metric and manifold data. In particular, but not exclusively, novel regression methods for specific metric quotient spaces are discussed. Special attention is given to handling discrete observations, since in practice curves and shapes are typically observed only as a discrete (often sparse or irregular) set of points. Similarly, density functions are usually not directly observed, but a (small) sample from the corresponding probability distribution is available. Overall, this work comprises six contributions that propose new methods for sparse functional object data and apply them to relevant real-world datasets, predominantly in a biomedical context

Description of Courses, 1979-80

Official publication of Cornell University V.71 1979/8