Eccfrog512ck2: An Enhanced 512-bit Weierstrass Elliptic Curve (2504.09584v2)

Abstract: Whilst many key exchange and digital signature methods use the NIST P256 (secp256r1) and secp256k1 curves, there is often a demand for increased security. With these curves, we have a 128-bit security. These security levels can be increased to 256-bit security with NIST P-521 Curve 448 and Brainpool-P512. This paper outlines a new curve - Eccfrog512ck2 - and which provides 256-bit security and enhanced performance over NIST P-521. Along with this, it has side-channel resistance and is designed to avoid weaknesses such as related to the MOV attack. It shows that Eccfrog512ck2 can have a 61.5% speed-up on scalar multiplication and a 33.3% speed-up on point generation over the NIST P-521 curve.

An Evaluation of Eccfrog512ck2: A Novel 512-bit Weierstrass Elliptic Curve

The paper presents Eccfrog512ck2, a newly developed 512-bit Weierstrass elliptic curve that aims to enhance cryptographic security while improving computational efficiency. This curve offers 256-bit security, presenting notable advantages compared to existing options such as NIST P-521. It introduces substantial improvements in performance, exemplified by up to a 61.5% speed-up in scalar multiplication and a 33.3% speed-up in point generation over the NIST P-521.

Key Features of Eccfrog512ck2

1. Security Enhancements: Eccfrog512ck2 operates within a 512-bit prime field that has been carefully selected to resist both classical and potential quantum cryptographic threats. The deterministic generation of coefficient $b$ through the cryptographically secure BLAKE3 hash function ensures both reproducibility and verifiable integrity.
2. Robustness Against Common Attacks: The curve design addresses vulnerabilities such as MOV and Twist attacks, with extensive validation to confirm resistance. This eliminates vulnerabilities associated with the reduction of elliptic curve problems to discrete logarithms, a threat notably addressed through the MOV attack.
3. Performance Optimization: Implementation incorporates advanced techniques such as wNAF, Montgomery Ladder, and GLV methods to optimize performance and provide side-channel attack resistance through constant-time operations and secure memory handling.

Methodological Insights

The design process emphasized the careful selection of the prime modulus and cryptographic parameters to address potential subgroup attacks. The curve is subject to extensive discriminant and order validations to ensure non-singularity and robustness. Comparative analyses were conducted on an AMD Ryzen 9 5950X processor, demonstrating the curve's efficiency in key cryptographic operations compared to established curves like NIST P-521.

Implications and Future Directions

The introduction of Eccfrog512ck2 presents significant implications for cryptographic practices, offering superior security and performance for applications requiring stringent cryptographic properties. This curve could serve as a viable alternative in blockchain and secure communication protocols, where enhanced security is paramount.

In terms of future developments, the curve's foundational security properties offer a promising basis for further exploration in quantum-resistant cryptographic methods. As cryptographic challenges evolve, this work sets the stage for advancing elliptic curve cryptography, providing insights into optimizing security and efficiency beyond conventional curves.

Overall, Eccfrog512ck2 offers a robust contribution to the elliptic curve cryptography domain, demonstrating the potential for enhanced security measures in increasingly complex cryptographic landscapes.