Abstract It is well-established that Shor’s algorithm can solve the discrete logarithm problem (DLP) in polynomial time. The hyperelliptic curve DLP (HCDLP) of genus 2 has found widespread industrial applications and remains an active research domain. In this work, we develop a quantum algorithm for solving HCDLP over binary fields $$\mathbb {F}_{2^n}$$ F 2 n by adapting Shor’s algorithmic framework. The core innovation lies in our divisor addition implementation, which combines the geometric interpretation of divisor operations with symmetric polynomial techniques. Using representative parameters ( $$n = 163, 283, 571$$ n = 163 , 283 , 571 ), we quantify the required quantum resources from the perspective of minimal qubit count, minimal T-gate usage, and minimal quantum depth. Furthermore, we compare the quantum resources required for solving HCDLP over binary fields with those for solving HCDLP over general prime fields and demonstrate the vulnerability of HCDLP-based cryptosystems to quantum attacks. Our analysis reveals that: (1) solving HCDLP over binary fields requires fewer quantum gates and less quantum depth compared to solving it over general prime fields; (2) the maximum achievable quantum depth for HCDLP attacks falls below NIST’s minimum security threshold of $$2^{40}$$ 2 40 for comparable protection levels, and (3) the quantum computational cost is orders of magnitude lower than the $$2^{157}$$ 2 157 resources needed for AES-128 attacks.

Abstract Verifiable delay functions (VDFs) and delay encryptions (DEs) are two important primitives in decentralized systems, while existing constructions are mainly based on time-lock puzzles. A disparate framework has been established by applying isogenies and pairings on elliptic curves. Following this line, we first employ Richelot isogenies and non-degenerate pairings from hyperelliptic curves for a new verifiable delay function, such that no auxiliary proof and interaction are needed for the verification. Then, we demonstrate that our scheme satisfies all security requirements, in particular, our VDF can resist several attacks, including the latest attacks for SIDH. Besides, resorting to the same techniques, a secure delay encryption from hyperelliptic curves is constructed by modifying Boneh and Frankiln’s IBE scheme, which shares the identical setup with our VDF scheme. As far as we know, these schemes are the first cryptographic applications from high-genus isogenies apart from basic protocols, i.e., hash functions and key exchange protocols.