Brace for impact: ECDLP challenges for quantum cryptanalysis

Abstract: Precise suites of benchmarks are required to assess the progress of early fault-tolerant quantum computers at economically impactful applications such as cryptanalysis. Appropriate challenges exist for factoring but those for elliptic curve cryptography are either too sparse or inadequate for standard applications of Shor's algorithm. We introduce a difficulty-graded suite of elliptic curve discrete logarithm (ECDLP) challenges that use Bitcoin's curve $y^{2}=x^{3}+7 \pmod p$ while incrementally lowering the prime field from 256 down to 6 bits. For each bit-length, we provide the prime, the base point and an example public key. All challenges are generated by a deterministic, reproducible procedure. We calibrate classical cost against Pollard's rho records and quantum cost against resource estimation results for Shor's algorithm. We compile Shor's ECDLP circuit to logical counts and map them to physical resources for various parameters of the surface code, the repetition cat code and the LDPC cat codes. Under explicit and testable assumptions on physical error rates, code distances, and non-Clifford supply, our scenarios place the full 256-bit instance within a 2027--2033 window. The challenge ladder thus offers a transparent ruler to track fault-tolerant progress on a cryptanalytic target of immediate relevance, and it motivates proactive migration of digital assets to post-quantum signatures.