3:00pm - 5:00pmApplications of finite fields theory
Chair(s): Antoine Joux (University of Sorbonne), Giacomo Micheli (EPFL), Violetta Weger (University of Zurich, Switzerland)
The theory of finite fields is one of the most important meeting points of Algebraic Geometry, Computer Science, and Number Theory. One of the most important challenges in the area is to develope the theory of finite fields in connection with useful applications, in particular in secure communication, coding theory, and pseudorandom number generation. In this minysimposium we plan to bring together experts from many different areas of the mathematics of communication who share the common interest towards the theory of finite fields. Our main purpose is to provide an overview of some of the cutting-edge research in the field, and to lay the fundations for new collaborations among researchers interested in applications of the theory of finite fields.
In the cryptographic setting, we focus on new post-quantum cryptographic schemes (Marco Baldi, Antoine Joux) and cryptanalysis (Gohar Kyureghyan, Yann Rotella). For pseudorandomness we propose construction of new pseudorandom generators (Federico Amadio Guidi, Laszlo Merai) and construction of polynomials over finite fields with given properties which are interesting for applications (Andrea Ferraguti).
(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)
Public key encryption and key exchange from LDPC codes: LEDAcrypt
Paolo Santini
Marche Polytechnic University
The pioneering work of McEliece in 1978 paved the way for code-based cryptography, which is still today a promising research area for the development of cryptographic primitives characterized by high efficiency and, most importantly, quantum resistance. Among several variants of the McEliece cryptosystem employing families of codes other than the original family of Goppa codes, those based on low-density parity-check (LDPC) codes have been shown able to achieve compact public keys and high algorithmic efficiency. This talk will recall the basic concepts of LDPC code-based cryptography, and then describe two primitives for asymmetric cryptography based on LDPC codes that are candidates to the NIST post-quantum cryptography standardization initiative: LEDAkem and LEDApkc.
Cryptological properties of mappings of finite fields
Gohar Kyureghyan
University of Rostock
Mappings used in some of cryptological primitives must be highly nonlinear, since linear ones are easy to predict. In this talk, we present several notions for optimal nonlinearity. We discuss connections between the different concepts and review known constructions and major open challenges in this research area.
Pseudorandom walks on elliptic curves
Laszlo Merai
RICAM
We give an overview of pseudorandom number generators (PRNGs) based on elliptic curves over finite fields. Many PRNGs are defined via a recursion law Pn = ψ(Pn-1) for some initial point P0 in E and a rational map (morphism) ψ:E → E of the curve E. An example for such PRNGs is the so-called power generator, where ψ is a scalar multiplication: ψ: P → eP for some integer e ≥ 2. We consider in detail the case when ψ is an arbitrary endomorphism of the curve.
We present bounds on the discrepancy and linear complexity of the obtained sequences.
Fractional Jumps and pseudorandom number generation
Federico Amadio Guidi
University of Oxford
In this talk we discuss a new construction of full orbit sequences in affine spaces over finite fields via Fractional Jumps of transitive projective automorphism, that is joint work with S. Lindqvist and G. Micheli. In dimension 1, our construction covers entirely the case of Inversive Congruential Generator (ICG) sequences. We explain how the sequences produced using Fractional Jumps enjoy the same discrepancy bounds as ICG sequences, but are less expensive to compute, thus representing a good source for pseudorandom number generation.