Victor Duarte Melo

Victor Duarte Melo

We introduce a mathematical and cryptographic framework for exact recovery of noisy hidden paths in high dimensional discrete path spaces. The work is inspired by the path integral viewpoint, where global quantities arise from contributions over many possible trajectories. Instead of approximating a global path sum, we study the inverse problem of recovering one exact hidden trajectory from incomplete, noisy, projected, and aggregated observables. The hidden object is a planted discrete path whose transitions may include macro steps, microscopic perturbations, and discrete noise. Public information is represented by large observable vectors rather than short hash digests, since excessive compression would bound the effective recovery problem by the digest size. We formalize several recovery notions, including planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings. The main distinction is that approximate reconstruction and exact recovery are fundamentally different tasks. A method may reveal coarse geometry or dominant regions without recovering the precise microscopic sequence defining the hidden path. We also discuss attack surfaces relevant to future cryptographic use, including linearization, lattice style recovery, dynamic programming, meet in the middle attacks, SAT and SMT formulations, approximation followed by rounding, witness collisions, and generic quantum search. This work does not claim a complete post quantum cryptosystem. It provides a formal framework for studying exact hidden path recovery as a possible foundation for future cryptographic constructions

Victor Duarte Melo

This paper presents ECCFROG522PP, a 522-bit prime-field elliptic curve in short Weierstrass form, designed with a focus on deterministic generation and public reproducibility. The central design principle is that all critical parameters are derived from a fixed public seed through a transparent and verifiable procedure. While many deployed systems rely on NIST P-256 and secp256k1, which target approximately 128-bit classical security, higher security applications typically consider curves such as NIST P-521, Curve448, and Brainpool P512. ECCFROG522PP is intended for the same general classical security range as P-521, with emphasis on transparency, auditability, and reproducibility rather than performance optimization. The curve parameters are generated through a BLAKE3-based deterministic pipeline with publicly specified indices. The resulting construction has prime order, cofactor one, and a deterministically derived base point of full order. The quadratic twist has a large proven prime factor, and the construction includes a documented lower bound on the embedding degree together with standard sanity checks against low embedding degree reductions and basic CM discriminant anomalies. The full generation and validation procedure can be reproduced end to end from public artifacts and reference scripts, enabling independent verification of all parameters and checks.

Victor Duarte Melo

HyperFrog is an experimental post-quantum Key Encapsulation Mechanism that explores a variant of the Learning With Errors (LWE) design space in which the secret is not sampled from an independent product distribution, but is deterministically derived from discrete topological structure. The scheme embeds a voxel grid in three dimensions and uses a topology mining procedure to search for connected subgraphs with prescribed complexity, measured by cyclomatic number (high genus). The resulting structure is encoded as a sparse binary secret vector, inducing strong geometric constraints on the secret distribution while retaining a large combinatorial search space. Encapsulation produces noisy linear relations over public parameters and derives the shared key via hashing; a Fujisaki-Okamoto style transform is used to target IND-CCA security in the random oracle model. We present the construction, parameterization, and serialization format, together with a reference implementation featuring self-tests and benchmarking on commodity CPUs. We also discuss how topology-derived secrets interact with known lattice and decoding attacks, and we outline open problems required for conservative parameter selection and for a full security analysis. HyperFrog is intended as a research vehicle rather than a production-ready KEM.