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<p>In today’s era, protecting information has become a paramount concern for both individuals and organizations. Ensuring the protection of sensitive data against unauthorized access is of utmost importance. When sharing data between different parties, it is essential to utilize secure communication channels. The only nonlinear element of a symmetric key encryption cipher that directly influences the security and efficiency of the encryption method is a substitution box (S-box). Therefore, producing S-boxes with outstanding performance and efficiency is desirable. Such structures are often constructed in algebra using Galois fields (GF). The local ring-based cryptosystem, which offers a significant security advantage against advanced cryptanalysis, is founded on the algebraic structure of local rings. In this article, we proposed a novel scheme that utilizes the local finite commutative chain ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(F_2\left[x\right]/\left\langle x^{10}\right\rangle={\mathbb{Z}}_2+u{\mathbb{Z}}_2+\cdots u^9{\mathbb{Z}}_2\)</EquationSource> </InlineEquation>, which consists of precisely <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\:1024\:\)</EquationSource> </InlineEquation>elements to generate highly nonlinear (NL) S-boxes. The use of an <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(10\times10\)</EquationSource> </InlineEquation>S-box is unfeasible due to memory limitations. To address this, we present an innovative approach by employing multiplicative group <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\:{U}_{{G}_{10}}\)</EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\:{R}_{10}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\:{R}_{10}\:\)</EquationSource> </InlineEquation>sub-module <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\:U\)</EquationSource> </InlineEquation> that contains all non-unit elements. Utilizing this concept, a four different <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(8\times8\)</EquationSource> </InlineEquation> S-boxes generation schemes are designed, which consider both unit and non-unit elements of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\:{R}_{10}\)</EquationSource> </InlineEquation>. This combination generates a large number of S-boxes (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\:{2}^{144}\)</EquationSource> </InlineEquation>) with NL 112 for most of them, resulting in an ideal value. Additionally, the proposed scheme is used for secure image encryption and obtains a much larger key space, with a total of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\:{2}^{436}\)</EquationSource> </InlineEquation>. This considerable increase in key space improves resilience to exhaustive search attacks, providing a greater level of encryption security. </p>

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Designing nonlinear component of block cipher over finite chain ring \(F_2\left[x\right]/\left\langle x^{10}\right\rangle\) and RGB image encryption

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • Huma Umbreen,
  • Hafeez Ur Rehman,
  • Tariq Shah

摘要

In today’s era, protecting information has become a paramount concern for both individuals and organizations. Ensuring the protection of sensitive data against unauthorized access is of utmost importance. When sharing data between different parties, it is essential to utilize secure communication channels. The only nonlinear element of a symmetric key encryption cipher that directly influences the security and efficiency of the encryption method is a substitution box (S-box). Therefore, producing S-boxes with outstanding performance and efficiency is desirable. Such structures are often constructed in algebra using Galois fields (GF). The local ring-based cryptosystem, which offers a significant security advantage against advanced cryptanalysis, is founded on the algebraic structure of local rings. In this article, we proposed a novel scheme that utilizes the local finite commutative chain ring \(F_2\left[x\right]/\left\langle x^{10}\right\rangle={\mathbb{Z}}_2+u{\mathbb{Z}}_2+\cdots u^9{\mathbb{Z}}_2\) , which consists of precisely \(\:1024\:\) elements to generate highly nonlinear (NL) S-boxes. The use of an \(10\times10\) S-box is unfeasible due to memory limitations. To address this, we present an innovative approach by employing multiplicative group \(\:{U}_{{G}_{10}}\) of \(\:{R}_{10}\) and \(\:{R}_{10}\:\) sub-module \(\:U\) that contains all non-unit elements. Utilizing this concept, a four different \(8\times8\) S-boxes generation schemes are designed, which consider both unit and non-unit elements of \(\:{R}_{10}\) . This combination generates a large number of S-boxes ( \(\:{2}^{144}\) ) with NL 112 for most of them, resulting in an ideal value. Additionally, the proposed scheme is used for secure image encryption and obtains a much larger key space, with a total of \(\:{2}^{436}\) . This considerable increase in key space improves resilience to exhaustive search attacks, providing a greater level of encryption security.