Paper 2026/1518

Practical Equivalent-Key Recovery in GRAFHEN

Abstract

GRAFHEN is a group-based homomorphic-encryption proposal whose public key is a rewriting system and whose secret key is a permutation representation. We present Maverick, an equivalent-key recovery attack. Maverick breaks every released GRAFHEN challenge, including the recommended two-copy $S_{11}$ instance: it reconstructs an equivalent key from public rules and correctly decrypts all $20{,}000$ supplied labelled ciphertexts. On $12$ independently generated recommended-parameter keys, the median end-to-end time is $552$ s and the median peak memory use is $11.28$ GB on an Apple M3 Pro. The attack converts selected public rewrite rules into group relators, reconstructs a regular action by Todd-Coxeter enumeration, recognizes the resulting permutation representations, and aligns the two sides through the public mixed relations. Public labelled encryptions then calibrate an equivalent decryptor. We prove a proof-carrying version of this procedure: a target-order table with a replayable trace certifies the recovered regular action. The reported $S_{11}$ experiments use target-order closure checks, complete-corpus verification, and decryptor validation. We also give an output-sensitive analysis and state the hypotheses needed to extrapolate beyond the measured instances. Following an independent key-recovery attack, GRAFHEN proposed in July 2026 to replace $S_{11}$ by $\mathrm{PSL}_2(343)$. We adapt Maverick to this setting and demonstrate complete public-rule recovery, alignment, certification, and calibration on generated $\mathrm{PSL}_2(169)$ instances. The two-side $q=169$ run decrypts all $5{,}000$ held-out ciphertexts in a $103$~s critical path using $0.93$ GiB peak memory; a $k=2$ admissibility-filtered corpus also succeeds. Under GRAFHEN's stated worst-case estimate for Dumezy's degree-based search, this degree $170$, $d=5$ setting already has cost $O(2^{850})$, well beyond the intended reach of that attack. At the target group, the post-enumeration recovery path from a generated regular action takes $35.0$ s and $1.92$ GiB peak memory. These results suggest that the proposed platform-group change is insufficient to rule out Maverick; recovery from an admissibility-filtered corpus at $q=343$ remains to be measured because the pre-filter KeyGen enumeration exceeded $32$ GiB of memory before it could produce a corpus.