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Construction of confusion component based on the isogeny of elliptic curves

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Abstract

The confusion-creating ability of a substitution box is indispensable. Its utmost involvement in the block cipher motivates researchers to investigate methods and construct cryptographically strong and efficient substitution boxes. In this paper, we define an isogeny that maps the elements of the base curve to the second curve. The points that can generate the maximal order subgroup of the elliptic curve points are the generators of the base curve. A group action is applied to the generators of the base curve to construct the initial S-boxes. The images of the generator points are established through isogeny and used to generate initial S-boxes from the second curve. The idea is cost-effective regarding computation, as both curves share the same prime field, and we only need to find a few generator points and the corresponding images. A suitable permutation from \(S_{n}\) is applied to the selected initial substitution box to improve the nonlinearity. The suggested S-box is balanced and highly nonlinear. We assessed the cryptographic strength of the suggested S-box against linear and differential probabilities. The smaller values of these tests ensure that the suggested S-box is cryptographically strong and can be used securely in any block cipher.

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  1. Institute of Space Technology, Islamabad, Pakistan

    Sadiqa Arshad

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Arshad, S. Construction of confusion component based on the isogeny of elliptic curves. Multimed Tools Appl 83, 47735–47749 (2024). https://doi.org/10.1007/s11042-023-17399-y

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