Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Computer Engineering and Sciences

First Advisor

Marius C. Silaghi

Second Advisor

Carlos Otero

Third Advisor

Susan Earles

Fourth Advisor

Eugene Dshalalow


Elliptic Curve Cryptography (ECC) has positioned itself as one of the most promising candidates for various applications since its introduction by Miller and Kolbitz in 1985 [53, 44]. The core operation for ECC is the scalar multiplication [k]P where many efforts have addressed its computation speed. Here we introduce an efficient approach for calculating elliptic curve operations by a novel regrouping of terms and creating new projective representation operators and increasing parallelism. These operators and the corresponding projective coordinate representations are shown to lead to adjusted versions of scalar multiplication algorithms that are evaluated. These techniques enable more opportunities for optimizing computations, directing to an important speed-up for every application based on elliptic curves such as encryption, crypt-analysis, digital signatures, and pseudo-random generators. Also benefiting from our work is the post-quantum cryptosystem, Supersingular Isogeny Diffie-Helman (SIDH). Its main weakness is elliptic curve computation complexity, that we improve, while its main quantum attacks complexity is maintained. For other elliptic curve schemes, the computation speed-up also favors attacks, which can however be compensated by increasing the size of the key. In addition, we simulate the modeled design as a hardware arithmetic circuit, to further quantify the improvements that can be obtained.