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<p>With the rapid development of computer networks and communication technologies, symmetric cryptographic primitives tailored for Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero-Knowledge (ZK) proofs have garnered significant attention from both industry and academia. In the design of cryptographic permutations and block ciphers, the partial substitution permutation network (P-SPN) framework, characterized by a nonlinear component that does not cover the entire state, has gained attention for its favorable implementation properties across scenarios. For word-oriented P-SPN structure schemes with a fixed linear layer, the choice of linear layer matrix critically influences their security: a weak linear layer matrix may enable an overly maximum invariant subspace that survives all rounds without activating any nonlinear operations. This study, with particular emphasis on linear layers composed of circulant block matrix with special block, investigates the maximum invariant subspace in P-SPN structure schemes where no S-boxes are activated. We first analyze the unique properties of such matrices. Next, we derive their minimal-degree annihilating polynomials and define the range of degrees for their minimal polynomials. Finally, using the annihilating polynomial degree as a basis, we establish a lower bound for the dimension of the maximum invariant subspace in P-SPN structure schemes employing these matrices. Notably, we precisely determine this dimension for P-SPN structure schemes with a single S-box in the nonlinear layer, and obtain a tighter lower bound for multi-S-box schemes under specific conditions. And we find that the dimensionality of the circulant block structures of such special matrix plays a crucial role in determining the dimensionality of the invariant subspace. The research results of this paper offer targeted guidance for designing linear layer matrices in P-SPN structure schemes, highlighting how component-specific differences impact security properties.</p>

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Security analysis of the P-SPN structure with a class of linear layer matrix against invariant subspace attack

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Ee Duan,
  • Wenling Wu

摘要

With the rapid development of computer networks and communication technologies, symmetric cryptographic primitives tailored for Multi-Party Computation (MPC), Fully Homomorphic Encryption (FHE), and Zero-Knowledge (ZK) proofs have garnered significant attention from both industry and academia. In the design of cryptographic permutations and block ciphers, the partial substitution permutation network (P-SPN) framework, characterized by a nonlinear component that does not cover the entire state, has gained attention for its favorable implementation properties across scenarios. For word-oriented P-SPN structure schemes with a fixed linear layer, the choice of linear layer matrix critically influences their security: a weak linear layer matrix may enable an overly maximum invariant subspace that survives all rounds without activating any nonlinear operations. This study, with particular emphasis on linear layers composed of circulant block matrix with special block, investigates the maximum invariant subspace in P-SPN structure schemes where no S-boxes are activated. We first analyze the unique properties of such matrices. Next, we derive their minimal-degree annihilating polynomials and define the range of degrees for their minimal polynomials. Finally, using the annihilating polynomial degree as a basis, we establish a lower bound for the dimension of the maximum invariant subspace in P-SPN structure schemes employing these matrices. Notably, we precisely determine this dimension for P-SPN structure schemes with a single S-box in the nonlinear layer, and obtain a tighter lower bound for multi-S-box schemes under specific conditions. And we find that the dimensionality of the circulant block structures of such special matrix plays a crucial role in determining the dimensionality of the invariant subspace. The research results of this paper offer targeted guidance for designing linear layer matrices in P-SPN structure schemes, highlighting how component-specific differences impact security properties.