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.

In this paper, we try to apply methods from topological data analysis (TDA) to study geometric properties invariant under code equivalence transformation, especially for the linear codes associated with almost perfect nonlinear (APN) functions which offer optimal resistance to differential attacks and are very important in the design of block ciphers in cryptography. By employing persistent homology from TDA and tools from graph theory, we present new CCZ-invariants for APN functions. Some of them are computationally efficient and sufficient to distinguish many known APN functions, includingx3andx9(resp.x33) over F27(resp. F29) for which previously known invariants fail to do so.

It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes$C$of packing radius$r$in$\mathbb {Z}^{n}$for$r\geq 2$and$n\geq 3$. In 2020, Leung and Zhou proved this conjecture for linear Lee codes with$r=2$. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius 2. Partial results was obtained recently by Xu and Zhou. In this paper, we show that if such codes exist in$\mathbb {Z}^{n}$, then$n$must belong to$\{1,2, 11, 29, 47, 56, 67, 79, 104, 121, 134, 191\}$.