Hardened CTIDH: Dummy-Free and Deterministic CTIDH

Abstract: Isogeny-based cryptography has emerged as a promising postquantum alternative, with CSIDH and its constant-time variants CTIDH and dCTIDH offering efficient group-action protocols. However, CTIDH and dCTIDH rely on dummy operations in differential addition chains (DACs) and Matryoshka, which can be exploitable by fault-injection attacks. In this work, we present the first dummy-free implementation of dCTIDH. Our approach combines two recent ideas: DACsHUND, which enforces equal-length DACs within each batch without padding, and a reformulated Matryoshka structure that removes dummy multiplications and validates all intermediate points. Our analysis shows that small primes such as 3, 5, and 7 severely restrict feasible DACsHUND configurations, motivating new parameter sets that exclude them. We implement dummy-free dCTIDH-2048-194 and dCTIDH-2048-205, achieving group action costs of roughly 357,000-362,000 Fp-multiplications, with median evaluation times of 1.59-1.60 (Gcyc). These results do not surpass dC-TIDH, but they outperform CTIDH by roughly 5% while eliminating dummy operations entirely. Compared to dCSIDH, our construction is more than 4x faster. To the best of our knowledge, this is the first efficient implementation of a CSIDH-like protocol that is simultaneously deterministic, constant-time, and fully dummy-free.