[Paper Review] Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

Natalia Tokareva, A. A. Gorodilova+6 more|arXiv (Cornell University)|Oct 16, 2017

Global Education Systems and Policies1 references3 citations

TL;DR

This paper presents mathematical solutions to 16 problems from the Third International Students' Olympiad in Cryptography (NSUCRYPTO’2016), covering advanced topics such as algebraic immune Boolean functions, big Fermat numbers, secret sharing schemes, pseudorandom binary sequences, biometric cryptosystems, and blockchain technology. A key contribution is the first-ever solution to an open problem in the Olympiad’s history, proposed by a participant during the competition.

ABSTRACT

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history.

Motivation & Objective

• To present and solve 16 complex mathematical problems from the NSUCRYPTO’2016 Olympiad, bridging pure mathematics and applied cryptography.
• To address unsolved research problems in mathematical cryptography, particularly in Boolean functions and number theory.
• To demonstrate the role of student participation in solving open problems, marking the first such instance in Olympiad history.
• To provide detailed solutions and insights into problems related to secret sharing, biometric systems, and pseudorandom sequences.
• To promote interdisciplinary research by integrating advanced mathematical techniques into cryptographic problem-solving frameworks.

Proposed method

• The Olympiad featured two rounds: a 4.5-hour individual round (Section A and B) and a one-week team-based research and programming round.
• Problems were drawn from diverse areas including algebraic immunity of vectorial Boolean functions, properties of big Fermat numbers, and cryptographic protocols.
• Solutions were derived using advanced mathematical tools such as finite field theory, combinatorial design, and number-theoretic analysis.
• The second round included two unsolved problems: one on algebraic immunity and another on big Fermat numbers, both of which were later solved by participants.
• Theoretical analysis and computational verification were used to validate solutions, especially for problems involving pseudorandom sequences and secret sharing schemes.
• A special prize was awarded for the first-ever participant-proposed solution to an open problem in the Olympiad’s history, highlighting the role of collaborative research.

Experimental results

Research questions

• RQ1 How can algebraic immunity be maximized in vectorial Boolean functions, and what are the structural constraints for such constructions?
• RQ2 What are the number-theoretic properties of big Fermat numbers, and how can they be leveraged in cryptographic primitives?
• RQ3 How can secret sharing schemes be designed to ensure information-theoretic security with minimal access structures?
• RQ4 What mathematical models underlie the construction of secure biometric cryptosystems and blockchain-based protocols?
• RQ5 Can open problems in mathematical cryptography be effectively solved through student-led research in competitive settings?

Key findings

• A participant successfully solved the open problem on algebraic immunity during the Olympiad, marking the first time in the event’s history that such a solution was achieved by a contestant.
• The solution to the algebraic immunity problem involved constructing a vectorial Boolean function with optimal resistance to algebraic attacks, verified through rigorous theoretical analysis.
• Several problems on pseudorandom binary sequences were solved using linear algebra and recurrence relation techniques, ensuring maximal period and balance properties.
• The big Fermat numbers problem was analyzed using modular arithmetic and properties of exponents, leading to a constructive proof of non-existence of certain solutions.
• Secret sharing schemes were successfully implemented using combinatorial designs and finite geometry, achieving threshold access structures with minimal redundancy.
• The Olympiad demonstrated that student researchers can contribute meaningful solutions to open problems in mathematical cryptography, especially when supported by structured, research-oriented competition formats.