VeloxReaper

Abstract

I present VeloxReaper, a memory-hard proof-of-work (PoW) function whose core arithmetic operates entirely in the negacyclic polynomial ring Rq = Zq [X]/(XN + 1) with N = 1024 and q = 12,289. Unlike existing memory-hard PoW schemes—Ethash, RandomX, Equihash— which compose classical hash functions with memory-filling graphs, VeloxReaper replaces the hash primitive with a lattice-native expansion function (LXOF) built from iterated NTT-domain squaring. I make three contributions. First, I give a complete specification of the construction and a formal game-based security model for memory-hard PoW (section 3). Second, I provide the first cryptographic analysis of the LXOF expansion function, proving that after k iterations the algebraic degree in the seed is 2k and that the output distribution has full min-entropy over Zq (section 5). I derive a rigorous time–memory trade-off lower bound S · T ≥ m2/4 via a red-blue pebbling argument on the bilinear DAG (section 7), and I prove entropy bounds on the τ -scrambler output that enable the subsequent SIS analysis (section 6). Third, I perform a corrected analysis of the SIS norm distribution (section 8), identifying a variance discrepancy in prior treatments and showing that the centered infinity norm of the Ring-SIS product concentrates near q/2 at the deployed parameters. I explain the design consequences: the SIS check serves as a structural algebraic certificate that the miner performed the full DAG computation, while fine-grained difficulty graduation is achieved by the preceding sequential lattice chain. I propose concrete parameter modifications for future deployments that would enable direct SIS-norm–based difficulty graduation (section 13). All numeric parameters stated in this paper (N , q, DAG size schedule, difficulty constants, etc.) are the production values used by the reference implementation