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This research proposes a unified framework to ground the hardness of cryptographic hash functions using algebraic graph theory and thermodynamics.

May 19, 2026Open Access

Physical Hardness of Cryptographic Hash Functions: A Unified Framework via Cayley Graph Expanders and Landauer Thermodynamics -- The B+C Conjecture Applied to YiJing Hash v2.0

Key Points

This research proposes a unified framework to ground the hardness of cryptographic hash functions using algebraic graph theory and thermodynamics.
Introduced the B+C Conjecture linking graph theory and thermodynamic physics.
Applied the conjecture to YiJing Hash v2.0 using Cayley graph expanders and Landauer thermodynamics.
Confirmed empirical properties of SL(2,F_5) Cayley graphs with Ramanujan characteristics.
Established a lower bound for algebraic complexity with Omega(1,431) matrix multiplications.
Showed that at least 1,370 steps are necessary for epsilon=2^{-128} statistical indistinguishability.
Demonstrated a Landauer weak bound of >= 7.35e-19 J for any physical inverter.

Abstract

We propose the B+C Conjecture: a unified framework grounding cryptographic hash hardness in algebraic graph theory (C: Cayley Graph Expanders) and thermodynamic physics (B: Landauer Principle). Applied to YiJing Hash v2. 0 (SL (2, F₁27), 8 Hou-Tian Ba-Gua generators): (C1) BSGS algebraic lower bound: Omega (1, 431) matrix multiplications. (C2) Expander Mixing Lemma: t >= 1, 370 steps for epsilon=2^-128 statistical indistinguishability. Empirically confirmed: SL (2, F₅) Cayley graph is Ramanujan (lambda₂/d=0. 809, exact computation). (B) Landauer weak bound (proven): any physical inverter dissipates >= 7. 35e-19 J. Independent of P vs NP. Landauer strong bound (open conjecture): thermal inaccessibility of erased information. Companion: YiJing Hash v1. 0 (doi: 10. 5281/zenodo. 20257027), v2. 0 (doi: 10. 5281/zenodo. 20257132).

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Yao-Kai Kao (Sun,) studied this question.

synapsesocial.com/papers/6a0bfe08166b51b53d379464 https://doi.org/https://doi.org/10.5281/zenodo.20257246

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