Igor Shparlinski Livres

Mathematics is presented as an essential tool for understanding the complexities of the modern world, where nonlinearities and feedback are prevalent. The book explores how various branches of mathematics, such as topology and logic, contribute significantly to other fields like physics and computer science. By applying a simple rewriting technique, the text illustrates the interconnectedness of mathematical disciplines and their practical applications, emphasizing their importance in both theoretical and applied contexts.

This book explores advanced techniques that establish rigorous lower bounds on the complexity of number-theoretic and cryptographic problems, focusing on pseudorandom properties of cryptographic primitives. It employs methods involving character sums and polynomial equations over finite fields, alongside sieve methods and lattice reduction algorithms. While the results are unconditionally proven and independent of conjectures, they are weaker than commonly accepted truths. The text also presents open problems and research proposals, emphasizing the significance of lower bounds in various mathematical functions and their implications for discrete logarithms.

Focusing on computational and algorithmic challenges in finite fields, this book explores polynomial factorization, the identification of irreducible and primitive polynomials, and their applications to elliptic curves. It also discusses recent advancements in congruences and computational number theory, covering topics like primality testing and modular arithmetic. The author highlights previously scattered results, including significant findings from Russian sources, making this work a comprehensive resource for applications in computer science, coding theory, and cryptography.

The book introduces new techniques which imply rigorous lower bounds on the complexity of some number theoretic and cryptographic problems. These methods and techniques are based on bounds of character sums and numbers of solutions of some polynomial equations over finite fields and residue rings. It also contains a number of open problems and proposals for further research.