Mathematical Hardness Assumptions in Modern Cryptography: A Comparative Survey from Classical Number Theory to Post‑Quantum Security

Yash Soni

Mathematical Hardness Assumptions in Modern Cryptography: A Comparative Survey from Classical Number Theory to Post‑Quantum Security

Author(s)	Yash Soni
Country	India
Abstract	Modern cryptography is based on the premise that certain mathematical problems are extremely difficult for computers to solve. This premise - the "hardness assumptions" - underpins secure communication on the internet, in banking, and the authentication of our digital selves. Traditional public-key cryptographic techniques such as RSA and Diffie-Hellman are built on number-theoretic problems such as factoring numbers into primes and discrete logarithms. They are considered to be "hard" for ordinary - "classical" - computers to solve. However, strong quantum algorithms reveal these to be easy problems for a sufficiently powerful large-scale quantum computer, thereby jeopardizing these systems. The secondary study examined principal mathematical hardness assumptions that underpin modern cryptography. We compared classical assumptions with post-quantum ones in post-quantum signature, key establishment, and encryption schemes such as code-based, hash-based, isogeny-based, lattice-based, and multivariate. We studied their theoretical and mathematical bases, security strengths, computational efficiencies, and defenses against quantum attacks, with specific attention devoted to NIST selection (Kyber, Dilithium, Falcon, and SPHINCS+). This analysis found that although classical assumptions are mature and widely used, they were susceptible to quantum attacks, while several of the post-quantum assumptions seem resistant but are not so mature and less well explored. We also discussed challenges that must be addressed when deploying and using the chosen post-quantum systems for infrastructure migration.
Keywords	Cryptography, Mathematical Hardness Assumptions, Computational Complexity, RSA, Discrete Logarithm Problem, Lattice‑Based Cryptography, Post‑Quantum Cryptography.
Field	Mathematics
Published In	Volume 8, Issue 5, September-October 2026
Published On	2026-10-04

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Mathematical Hardness Assumptions in Modern Cryptography: A Comparative Survey from Classical Number Theory to Post‑Quantum Security

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