Francisco Javier Lobillo

A ROM IND-CCA2 Key Encapsulation Mechanism (KEM) supported by a Public Key Encryption (PKE) scheme based on the Weighted Modular Subset Sum Problem (WMSSP) is proposed. Concretely, it is proven that for certain sets of integers, whose inverses modulo a large predetermined integer have bounded bit size, an algorithm based on a truncated version of the extended Euclidean algorithm can effectively solve the WMSSP. The security analysis is conducted through the connection established between the Binary Subset Sum Problem and the Shortest Vector Problem. Various attacks against our proposal, relying on the computation of a shortest vector, have been analyzed. It is widely accepted that the problem supporting the security of this cryptosystem exhibits resilience against quantum threats. Moreover, the key generation and encryption-decryption processes exhibit remarkable time efficiency, as they rely exclusively on basic integer arithmetic.

We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.

Linear complementary pairs (LCPs) of codes have been studied since they were introduced in the context of discussing mitigation measures against possible hardware attacks to integrated circuits. In this situation, the security parameters for LCPs of codes are defined as the (Hamming) distance and the dual distance of the codes in the pair. We study the properties of LCPs of skew constacyclic codes, since their algebraic structure provides tools for studying their duals and their distances. As a result, we give a characterization for those pairs, as well as multiple results that lead to constructing pairs with designed security parameters. We extend skew BCH codes to a constacyclic context and show that an LCP of codes can be immediately constructed from a skew BCH constacyclic code. Additionally, we describe a Hamming weight-preserving automorphism group in the set of skew constacyclic codes, which can be used for constructing LCPs of codes.

A class of linear codes that extends classical Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.

We design a decoding algorithm for linear codes over finite chain rings given by their parity check matrices. It is assumed that decoding algorithms over the residue field are known at each degree of the adic decomposition.

This note does not concern itself with theoretical aspects of quantum groups. A good introduction on this topic for Ring Theorists is [27]. Here, we will show that some effective methods stemming from Commutative Algebra could be applied to most relevant quantum groups.