Acceleration of arithmetic computations in elliptic curve cryptography

Abstract

The Elliptic Curve Digital Signature Algorithm (ECDSA) is a fundamental cryptographic mechanism for ensuring the authenticity and integrity of digital communications. A central operation in signature verification is double point multiplication (DPM), whose computational structure directly affects performance, memory consumption, and resistance to side-channel attacks (SCAs). This thesis proposes simple and uniform constant-time algorithms for DPM based on an iterative left-to-right windowing method that performs simultaneous recoding and evaluation in a single pass. This design improves efficiency, reduces memory requirements, and strengthens protection against timing and power-analysis attacks. The proposed methods are analyzed through precise analytic formulas covering speed, memory, and security, and are compared with state-of-the-art approaches over NISTrecommended curves, as well as twisted-Edwards and Montgomery models. Unlike curve-specific techniques, the proposed algorithms are field-independent and flexible, enabling practical trade-offs between speed, memory, and security. When applied to ECDSA, the algorithms reduce point additions without increasing point doublings and require minimal precomputation, resulting in significant computational savings compared to existing methods