Improvement of the square-root Vélu's formulas for isogeny-based cryptography

Isogeny-based cryptography is a promising candidate for post-quantum cryptography. Speeding up the isogeny computation of large prime degrees up to several thousands is a main research topic in isogeny-based cryptography. The currently most efficient method for computing isogenies is the square-root Vélu's formulas. In this paper, we propose an efficient variant of the square-root Vélu's formulas by introducing an optimal index system using a redundant integer representation. Our index system improves the isogeny computation in 44% of 367 primes between 97 and 2689 and is approximately 6.6% faster for primes 1279 and 2687.