Avoiding trusted setup in isogeny-based commitments

The article delves into the critical role of commitment schemes in cryptography, particularly in proofs of knowledge and electronic voting. It highlights the vulnerabilities of the widely used Pedersen commitment scheme, which is not quantum-safe due to its reliance on the discrete logarithm problem. The focus then shifts to isogeny-based commitments, which offer a promising alternative for post-quantum cryptography. The existing isogeny-based commitment scheme by Sterner is examined, revealing its dependence on a trusted setup to ensure security. The article introduces two innovative approaches to eliminate the need for a trusted setup. The first approach utilizes a specific starting curve that avoids endomorphisms of small degree, resulting in a perfectly binding and computationally hiding commitment scheme. The second approach employs a uniformly random supersingular elliptic curve, ensuring that any endomorphisms of small degree are hard to compute, leading to a computationally binding and hiding scheme. The article provides a comprehensive comparison of these approaches with the original Sterner scheme, emphasizing their security properties and efficiency gains. It concludes by discussing the implications of these findings for the future development of isogeny-based cryptographic protocols.