Author Type

Graduate Student

Date of Award

Fall 11-16-2025

Document Type

Dissertation

Publication Status

Version of Record

Submission Date

November 2025

Department

Mathematical Sciences

Degree Name

Doctor of Philosophy (PhD)

Thesis/Dissertation Advisor [Chair]

Dipayan Das

Thesis/Dissertation Co-Chair

Edoardo Persichetti

Abstract

Classical cryptographic schemes, which are based on the hardness of factorization and discrete logarithmic problems, can be efficiently solved by the Shor algorithm on a quantum computer. This motivated the 2016 National Institute of Standards and Technology (NIST) call to identify efficient and secure cryptographic schemes that are resilient to potential attacks from both classical and quantum adversaries, a field referred to as post-quantum cryptography. In this work, we focus on designing efficient post-quantum cryptographic primitives based on code-based and lattice-based assumptions, and we analyze their underlying hardness using quantum cryptanalysis. On the construction side, we propose a ring signature scheme and an identity-based signature scheme based on the Code Equivalence Problem, leveraging the LESS identification scheme and the Calamari-Falafl framework. The proposed ring signature achieves small public keys (11.57 kB), and its signature size grows logarithmically with the number of users in the ring, outperforming existing code-based solutions while remaining competitive with other post-quantum schemes, particularly for large ring sizes. Furthermore, we introduce cryptographic schemes based on the Module NTRU problem, a generalization of the NTRU problem that provides better flexibility in parameter selection. Building on this, we design compact encryption schemes that achieve a low decryption failure rate, with a proposed parameter set offering the smallest ciphertext size among NIST Level 3 security schemes. Additionally, we present the design of signature schemes, one of which achieves the smallest provably secure signature size in the Quantum Random Oracle Model (QROM). On the cryptanalysis side, we present a concrete quantum resource estimation for lattice enumeration based on Montanaro’s algorithm, together with a detailed implementation in the quantum circuit model, and show how to optimize the depth of the circuit through parallelized design components. The second contribution is a quasi-polynomial-time algorithm for the Extrapolated Dihedral Coset Problem (EDCP) over power-of-two moduli. Although our results on EDCP do not compromise the security of LWE with standard parameters, they offer insight into the complexity of LWE.

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