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The HyperFrog Cryptosystem: High-Genus Voxel Topology as a Trapdoor for Post-Quantum KEMs

Victor Duarte Melo

Abstract

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.

The HyperFrog Cryptosystem: High-Genus Voxel Topology as a Trapdoor for Post-Quantum KEMs

Abstract

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.
Paper Structure (15 sections, 4 equations, 5 figures, 6 tables)

This paper contains 15 sections, 4 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Revision summary and statement of changes
  3. Design overview and parameter set
  4. Graph model and topological statistic
  5. Formal connected graph miner
  6. Formal and engineering miner separation
  7. HyperFrog KEM construction
  8. Correctness analysis
  9. Security discussion and scope of claims
  10. Implementation details in hyperfrog36.cpp
  11. Experimental methodology
  12. Benchmark results
  13. Discussion of the benchmark numbers
  14. Limitations and open problems
  15. Conclusion

Figures (5)

  • Figure 1: Example voxel rendering used as a qualitative illustration of the kind of connected occupancy field produced by the formal connected growth miner. The formal invariant mined by the implementation is graph cycle rank $\beta_1$ of the occupied six neighbor graph.
  • Figure 2: Representative formal KEM latency distribution from the 16 thread, 200 iteration comparison run. Decapsulation is consistently slower than encapsulation, but the spread remains compact.
  • Figure 3: Observed mean timings in the formal thread sweep. The present miner is so light in this regime that 32 threads add overhead rather than improving performance.
  • Figure 4: Observed geometry in the formal and practical comparison runs. The formal miner remains on exact weight 2048 with much higher cycle rank, while the practical miner explores a distinctly different region.
  • Figure 5: File processing breakdown across the uploaded formal file benchmarks. The dominant fixed cost is password based secret key unlock. The cryptographic file path then scales with file size.