Complementary information set (CIS) codes and higher-order complementary information set codes were introduced and studied by Carlet, Gaborit, Kim, Solé, Freibert, Guilley, Kiermaier in two papers of IEEE Transactions on Information Theory. These codes are closely connected to correlation-immune vectorial Boolean functions in the security of hardware implementations of cryptographic primitives and can be used to improve the cost of masking cryptographic algorithms against side channel attacks. However there is no infinite family of optimal CIS codes or optimal higher-order CIS codes reported in the literature. In this paper, we construct infinitely many infinite families of optimal binary and p-ary higher-order CIS codes. Some of them are Griesmer codes or close to the Griesmer bound. Many optimal or best known CIS codes or t-CIS codes over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{F}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{F}_3,\textbf{F}_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">F</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="bold">F</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{F}_7$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">F</mi> <mn>7</mn> </msub> </math></EquationSource> </InlineEquation> then are derived. Several infinite families of explicit binary or p-ary CIS <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$[2n,n]_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">[</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> codes, with a square-root-like lower bound on the minimum distance or a minimum distance of at least <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{2n}{\log 2n}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mo>log</mo> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </math></EquationSource> </InlineEquation>, are also given. The application in cryptography for constructing correlation-immune functions and correlation-immune t-tuples is discussed.

Optimal CIS codes, optimal t-CIS codes and their applications in cryptography

期刊论文
发表日期：

2025年12月24日

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Hao Chen,
Shengwei Liu,
Conghui Xie,
Hongwei Liu

摘要

Complementary information set (CIS) codes and higher-order complementary information set codes were introduced and studied by Carlet, Gaborit, Kim, Solé, Freibert, Guilley, Kiermaier in two papers of IEEE Transactions on Information Theory. These codes are closely connected to correlation-immune vectorial Boolean functions in the security of hardware implementations of cryptographic primitives and can be used to improve the cost of masking cryptographic algorithms against side channel attacks. However there is no infinite family of optimal CIS codes or optimal higher-order CIS codes reported in the literature. In this paper, we construct infinitely many infinite families of optimal binary and p-ary higher-order CIS codes. Some of them are Griesmer codes or close to the Griesmer bound. Many optimal or best known CIS codes or t-CIS codes over $\textbf{F}_2$ $F_{2}$ , $\textbf{F}_3,\textbf{F}_5$ $F_{3}, F_{5}$ and $\textbf{F}_7$ $F_{7}$ then are derived. Several infinite families of explicit binary or p-ary CIS $[2n,n]_p$ ${[2 n, n]}_{p}$ codes, with a square-root-like lower bound on the minimum distance or a minimum distance of at least $\frac{2n}{\log 2n}$ $\frac{2 n}{log 2 n}$ , are also given. The application in cryptography for constructing correlation-immune functions and correlation-immune t-tuples is discussed.