Code-based masking (CBM) is a recent line of countermeasures for protecting cryptographic implementations, in which the underlying linear codes play a critical role in its resilience against side-channel attacks. However, constructing linear codes that can have the best side-channel protection is a nontrivial task. The recent work shows that the central problem is to optimize the dual distance <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d^\bot $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>d</mi> <mi>⊥</mi> </msup> </math></EquationSource> </InlineEquation> and the kissing number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_{d^\bot }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <msup> <mi>d</mi> <mi>⊥</mi> </msup> </msub> </math></EquationSource> </InlineEquation> of the corresponding linear code. It turns out that finding optimal linear codes for CBMs will have exponential complexity as the order increases. In this paper, we propose two efficient constructions of the optimal linear code for CBMs, particularly for the SSS (Shamir’s secret sharing)-based masking and the inner product masking (IPM). We demonstrate all the optimal (n, t)-SSS-based masking with the largest dual distance and the smallest kissing number for various pairs of number of shares n and the security order t. Furthermore, we compute the dual distance and the kissing number of the generalized SSS (GSSS)-based maskings. We derive an optimal (4, 2)-GSSS-based masking with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(d^\bot ,B_{d^\bot })=(7,38)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>d</mi> <mi>⊥</mi> </msup> <mo>,</mo> <msub> <mi>B</mi> <msup> <mi>d</mi> <mi>⊥</mi> </msup> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo>,</mo> <mn>38</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, while the optimal (4, 2)-SSS-based masking only gives a dual distance up to 6. As for IPM, we use Kronecker product to construct CBMs at higher orders. We derive several instances of IPM with large dual distances at very high orders.

Constructing optimal linear codes for code-based masking schemes of higher-orders

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发表日期：

2026年3月26日

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Jihao Fan,
Wei Cheng,
Yongbin Zhou,
Sylvain Guilley

摘要

Code-based masking (CBM) is a recent line of countermeasures for protecting cryptographic implementations, in which the underlying linear codes play a critical role in its resilience against side-channel attacks. However, constructing linear codes that can have the best side-channel protection is a nontrivial task. The recent work shows that the central problem is to optimize the dual distance $d^\bot $ $d^{⊥}$ and the kissing number $B_{d^\bot }$ $B_{d^{⊥}}$ of the corresponding linear code. It turns out that finding optimal linear codes for CBMs will have exponential complexity as the order increases. In this paper, we propose two efficient constructions of the optimal linear code for CBMs, particularly for the SSS (Shamir’s secret sharing)-based masking and the inner product masking (IPM). We demonstrate all the optimal (n, t)-SSS-based masking with the largest dual distance and the smallest kissing number for various pairs of number of shares n and the security order t. Furthermore, we compute the dual distance and the kissing number of the generalized SSS (GSSS)-based maskings. We derive an optimal (4, 2)-GSSS-based masking with $(d^\bot ,B_{d^\bot })=(7,38)$ $(d^{⊥}, B_{d^{⊥}}) = (7, 38)$ , while the optimal (4, 2)-SSS-based masking only gives a dual distance up to 6. As for IPM, we use Kronecker product to construct CBMs at higher orders. We derive several instances of IPM with large dual distances at very high orders.