Some Algebraic and Number Theoretic Aspects of Classical and Post Quantum Cryptography

The thesis focuses on some of the mathematical aspects in the perspective of contemporary issues in information security. In the paradigm of post-quantum cryptography, the design of a new digital signature scheme is discussed, which is based on isogenies of supersingular elliptic curves defined over a finite field. In classical cryptography, a novel geometric interpretation of the elliptic curve discrete logarithm problem is illustrated. Additionally, the application of recursive towers for producing high order elements in finite fields is studied. Another concept which is explored in the thesis is estimation of class numbers, which is necessary for designing secure cryptosystems over number fields.