Paper 2025/1731

ECCFROG522PP: An Enhanced 522-bit Weierstrass Elliptic Curve

Abstract

Whilst many key exchange and digital signature systems still rely on NIST P-256 (secp256r1) and secp256k1, offering around 128-bit security, there is an increasing demand for transparent and reproducible curves at the 256-bit security level. Standard higher-security options include NIST P-521, Curve448, and Brainpool-P512. This paper presents ECCFROG522PP ('Presunto Powered'), a 522-bit prime-field elliptic curve that delivers security in the same classical $\sim$260-bit ballpark as NIST P-521, but with a fundamentally different design philosophy. All of the curve parameters are deterministically derived from a fixed public seed via BLAKE3, with zero hidden choices. The curve has prime order (cofactor = 1), a verified twist with a proven approximate 505-bit prime factor, safe embedding degree, and passes anti-MOV checks up to k <200 and CM discriminant sanity up to 100k. Unlike prior opaque or ad-hoc constructions, ECCFROG522PP is fully reproducible: anyone can regenerate and verify it byte-for-byte using the published scripts. The intent is not to outperform NIST P-521 in raw speed, but to maximise trust, verifiability, and long-term auditability in a practical curve of equivalent security level.

Note: ECCFrog512CK2 was originally defined over a 512-bit prime field (https://eprint.iacr.org/2025/660), with parameters derived through iterative testing using BLAKE3. Although it achieved prime order and twist security, later analysis revealed weaknesses in its structure that limited its reliability for long-term use. ECCFrog522PP was designed as a direct response to these issues. Built on a 522-bit prime field, it eliminates the flaws found in ECCFrog512CK2 by adopting a stronger foundation, an enhanced security margin, and a fully transparent, reproducible parameter-generation process. Under the “Presunto Power” philosophy, ECCFrog522PP provides both cryptographic soundness and auditability, ensuring that no hidden weaknesses remain. As a result, ECCFrog512CK2 is considered a stepping stone in the development process, while ECCFrog522PP stands as the corrected and trustworthy curve.