Cryptographic Hardness Assumptions Based on Number-Theoretic Problems

This paper explores the role of number theoretic methods in cryptography, emphasizing the use of primitive Pythagorean triples for developing new encryption and decryption algorithms through a fundamental Pythagorean tree. It reviews computational number theory, focusing on primality testing and various tests like Miller-Rabin and elliptic curve, which enhance algorithm efficiency. The discussion includes references to works by Silverman and Tate, and Hankerson et al., highlighting implementation challenges in elliptic-curve cryptography. The paper asserts the importance of number theoretic approaches and critiques existing algorithms for their speed and security limitations, noting ongoing research efforts toward improved solutions.