Date of Award
2024
Document Type
Dissertation
Publication Status
Version of Record
Submission Date
October 2026
Thesis/Dissertation Advisor [Chair]
Persichetti, Edoardo
Thesis/Dissertation Co-Chair
Karabina, Koray
Abstract
An efficient scalar multiplication algorithm is vital for elliptic curve cryptosystems. The first part of this dissertation focuses on a scalar multiplication algorithm based on scalar recodings resistant to timing attacks. The algorithm utilizes two recoding methods: Recode, which generalizes the non-zero signed all-bit set recoding, and Align, which generalizes the sign aligned columns recoding. For an ℓ-bit scalar split into d subscalars, our algorithm has a computational cost of ⌈⌈ℓ logk(2)⌉/d⌉ point additions and k-scalar multiplications and a storage cost of kd−1(k − 1) – 1 points on E. The “split and comb” method further optimizes computational and storage complexity. We find the best setting to be with a fixed base point on a Twisted Edwards curve using a mix of projective and extended coordinates, with k = 2 generally offering the best performance. However, k = 3 may be better in certain applications. The second part of this dissertation is dedicated to constant-time polynomial inversion algorithms in Post-Quantum Cryptography (PQC). The computation of the inverse of a polynomial over a quotient ring or finite field is crucial for key generation in post-quantum cryptosystems like NTRU, BIKE, and LEDACrypt. Efficient algorithms must run in constant time to prevent side-channel attacks. We examine constant-time algorithms based on Fermat’s Little Theorem and the Extended GCD Algorithm, providing detailed time complexity analysis. We find that the constant-time Extended GCD inversion algorithm is more efficient, performing fewer field multiplications. Additionally, we explore other exponentiation algorithms similar to the Itoh-Tsuji inversion method, which optimizes polynomial multiplications in the BIKE/LEDACrypt setup. Recent results on hardware implementations are also discussed.
Extent
122 p.
Recommended Citation
Dutta, Abhraneel, "ANALYSIS OF CRYPTOGRAPHIC EFFICIENCY: ELLIPTIC CURVE SCALAR MULTIPLICATION AND CONSTANT-TIME POLYNOMIAL INVERSION IN POST-QUANTUM CRYPTOGRAPHY" (2024). Electronic Theses and Dissertations 2020 - Present. 1361.
https://digitalcommons.fau.edu/etd_general/1361
Identifier
FA00014492
Comments
Includes bibliography.|Dissertation (PhD)--Florida Atlantic University, 2024.|FAU Electronic Theses and Dissertations Collection|Includes bibliography.