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In this paper, we propose PoC Hash, a post-quantum cryptographic hash function that operates natively in Hamilton’s quaternion algebra over a finite field, with security based on the hardness of the Multivariate Cubic (MC) problem. Each compression step takes two input quaternions, forms their non-abelian commutator, multiplies the result by the conjugate of a third quaternion, raises the product to the third power, and finally adds a fixed non-central offset. This procedure costs only five quaternion multiplications (roughly 80 finite-field multiplies), so software performance is expected to be comparable to the SHA-2 family while scaling efficiently on SIMD and GPU hardware. Finding a pre-image for a t-block message reduces to solving an \((n,m,q)=(6t,4t,q\approx 2^{128})\) multivariate cubic system. PoC Hash thus combines prime-field efficiency with a non-commutative hardness foundation and meets the 256-bit security goal.

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PoC Hash: A Power-of-Commutators Hash over Hamiltonian Quaternions

  • 会议论文

  • 发表日期：

    2026年

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  • Hyungrok Jo,
  • Junji Shikata

摘要

In this paper, we propose PoC Hash, a post-quantum cryptographic hash function that operates natively in Hamilton’s quaternion algebra over a finite field, with security based on the hardness of the Multivariate Cubic (MC) problem. Each compression step takes two input quaternions, forms their non-abelian commutator, multiplies the result by the conjugate of a third quaternion, raises the product to the third power, and finally adds a fixed non-central offset. This procedure costs only five quaternion multiplications (roughly 80 finite-field multiplies), so software performance is expected to be comparable to the SHA-2 family while scaling efficiently on SIMD and GPU hardware. Finding a pre-image for a t-block message reduces to solving an \((n,m,q)=(6t,4t,q\approx 2^{128})\) multivariate cubic system. PoC Hash thus combines prime-field efficiency with a non-commutative hardness foundation and meets the 256-bit security goal.