Construction of $${\text {MDS}}$$ matrices from generalized Feistel structures

Autor: Mahdi Sajadieh, Mohsen Mousavi
Rok vydání: 2021
Předmět:
Zdroj: Designs, Codes and Cryptography. 89:1433-1452
ISSN: 1573-7586
0925-1022
DOI: 10.1007/s10623-021-00876-6
Popis: This paper investigates the construction of $${\text {MDS}}$$ matrices with generalized Feistel structures ( $${\text {GFS}}$$ ). The approach developed by this paper consists in deriving $${\text {MDS}}$$ matrices from the product of several sparser matrices. This can be seen as a generalization to several matrices of the recursive construction which derives $${\text {MDS}}$$ matrices as the powers of a single companion matrix. In other words, the idea of this paper is to explore a space of matrices with a $${\text {GFS}}$$ structure, and then to search for instantiations of the binary linear functions so that the resulting matrix is both $${\text {MDS}}$$ and efficient to implement with respect to the number of $${\text {XOR}}$$ gates and the depth of the circuit. In this direction we first give some theoretical results on the iteration of $${\text {GFS}}$$ . We then using $${\text {GFS}}$$ with minimal diffusion rounds, propose some types of sparse matrices that are called extended primitive $${\text {GFS}}$$ ( $${\text {EGFS}}$$ ) matrices. Next, by applying binary linear functions to several round of $${\text {EGFS}}$$ matrices, we introduce lightweight $$4\times 4$$ , $$6\times 6$$ and $$8\times 8$$ $${\text {MDS}}$$ matrices that are implemented with 67, 156 and 260 $${\text {XOR}}$$ over 8-bit input, respectively. The results match the best known lightweight $$4\times 4$$ $${\text {MDS}}$$ matrix and improve the best known $$6\times 6$$ and $$8\times 8$$ $${\text {MDS}}$$ matrices. Moreover, we propose $$8\times 8$$ Near- $${\text {MDS}}$$ matrices such that the implementation cost of the proposed matrices are 108 and 204 $${\text {XOR}}$$ over 4-bit and 8-bit inputs, respectively. On the whole, the construction presented in this paper is relatively general and can be applied for other matrix dimensions and finite fields as well.
Databáze: OpenAIRE
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