[Paper Review] A novel encryption algorithm using multiple semifield S-boxes based on permutation of symmetric group

Iqtadar Hussain, Amir Anees+2 more|arXiv (Cornell University)|Apr 26, 2020

Chaos-based Image/Signal Encryption21 references4 citations

TL;DR

This paper proposes a novel block cipher using multiple semifield-based S-boxes generated via permutation of the symmetric group $S_8$, enhancing cryptographic strength through algebraic robustness. By applying $S_8$ permutations to 12,781 non-equivalent semifield S-boxes from Dumas-Orfila (2004), the method generates 515,329,920 new S-boxes with identical cryptographic properties, enabling a secure, flexible, and efficient encryption algorithm resistant to linear and differential cryptanalysis with $LP_{max} = 2^{-1077}$ and $DP_{max} = 2^{-4.05}$.

ABSTRACT

With the tremendous benefits of internet and advanced communications, there is a serious threat from the data security perspective. There is a need of secure and robust encryption algorithm that can be implemented on each and diverse software and hardware platforms. Also, in block symmetric encryption algorithms, substitution boxes are the most vital part. In this paper, we investigate semifield substitution boxes using permutation of symmetric group on a set of size 8 S_8 and establish an effective procedure for generating S_8 semifield substitution boxes having same algebraic properties. Further, the strength analysis of the generated substitution boxes is carried out using the well-known standards namely bijectivity, nonlinearity, strict avalanche criterion, bit independence criterion, XOR table and differential invariant. Based on the analysis results, it is shown that the cryptographic strength of generated substitution boxes is on par with the best known $8 imes 8$ substitution boxes. As application, an encryption algorithm is proposed that can be employed to strengthen any kind of secure communication. The presented algorithm is mainly based on the Shannon idea of (S-P) network where the process of substitution is performed by the proposed S_8 semifield substitution boxes and permutation operation is performed by the binary cyclic shift of substitution box transformed data. In addition, the proposed encryption algorithm utilizes two different chaotic maps. In order to ensure the appropriate utilization of these chaotic maps, we carry out in-depth analyses of their behavior in the context of secure communication and apply the pseudo-random sequences of chaotic maps in the proposed image encryption algorithm accordingly. The statistical and simulation results imply that our encryption scheme is secure against different attacks and can resist linear and differential cryptanalysis.

Motivation & Objective

• To address potential vulnerabilities in finite field-based S-boxes from unknown algebraic attacks by leveraging the non-associative structure of semifields.
• To generate a large number of high-quality, non-equivalent S-boxes with consistent cryptographic properties using symmetric group permutations.
• To design a flexible, secure block cipher suitable for low-profile and diverse hardware/software platforms.
• To integrate chaotic maps for dynamic key scheduling and enhanced diffusion in the encryption process.
• To validate the proposed algorithm’s resistance against standard cryptanalytic attacks through statistical and simulation analysis.

Proposed method

• The method generates $S_8$ semifield S-boxes by applying the symmetric group $S_8$ to each of the 12,781 non-equivalent semifield S-boxes from Dumas-Orfila (2004), producing $8! = 40320$ permutations per base S-box.
• Each S-box is constructed over a pseudo-extended semifield of order $2^4$, with addition and multiplication defined using base field operations and specific algebraic rules.
• The encryption algorithm employs a substitution-permutation network (S-P) structure with three modules: substitution using multiple S-boxes, binary cyclic shift-based permutation, and XOR whitening.
• Two distinct chaotic maps are used to generate pseudo-random sequences that control the key scheduling and transformation processes, enhancing diffusion and confusion.
• The algorithm is designed for 6 rounds, with each module applied using sequences derived from the respective chaotic map to ensure dynamic and unpredictable behavior.
• Cryptographic strength is evaluated using standard metrics: bijectivity, nonlinearity, strict avalanche criterion (SAC), bit independence criterion (BIC), XOR table, and differential uniformity.

Experimental results

Research questions

• RQ1 Can semifield-based S-boxes generated via symmetric group permutations achieve cryptographic strength comparable to the best-known $8\times8$ S-boxes?
• RQ2 Does the use of non-associative semifield structures improve resistance against unknown algebraic attacks compared to finite field-based S-boxes?
• RQ3 Can the symmetric group $S_8$ be effectively used to generate a large number of non-equivalent, high-quality S-boxes from a small base set of semifield S-boxes?
• RQ4 How does the integration of two chaotic maps enhance the security and flexibility of the proposed block cipher?
• RQ5 To what extent does the proposed algorithm resist linear and differential cryptanalysis, as measured by $LP_{max}$ and $DP_{max}$?

Key findings

• The proposed method generates 515,329,920 new S-boxes from 12,781 base semifield S-boxes using $S_8$ permutations, all preserving identical cryptographic properties.
• The maximum linear approximation probability ($LP_{max}$) of the 4-round algorithm with 256 active S-boxes is $2^{-1077}$, indicating strong resistance to linear cryptanalysis.
• The maximum differential uniformity ($DP_{max}$) of the S-boxes used is $2^{-4.05}$, demonstrating robustness against differential cryptanalysis.
• The algorithm achieves high nonlinearity and satisfies strict avalanche criterion (SAC) and bit independence criterion (BIC), confirming strong confusion and diffusion properties.
• Statistical and simulation results confirm that the proposed encryption scheme is secure against known attacks and remains flexible for adaptation in terms of rounds and S-box count.
• The integration of two chaotic maps ensures dynamic key scheduling and enhances resistance to known cryptanalytic techniques, supporting secure communication applications.