Abstract
The paper investigates the maximum distance separable (MDS) matrix over the matrix polynomial residue ring. Firstly, by analyzing the minimal polynomials of binary matrices with 1 XOR count and element-matrices with few XOR counts, we present an efficient method for constructing MDS matrices with as few XOR counts as possible. Comparing with previous constructions, our corresponding constructions only cost 1 minute 27 seconds to 7 minutes, while previous constructions cost 3 days to 4 weeks. Secondly, we discuss the existence of several types of involutory MDS matrices and propose an efficient necessary-and-sufficient condition for identifying a Hadamard matrix being involutory. According to the condition, each involutory Hadamard matrix over a polynomial residue ring can be accurately and efficiently searched. Furthermore, we devise an efficient algorithm for constructing involutory Hadamard MDS matrices with as few XOR counts as possible. We obtain many new involutory Hadamard MDS matrices with much fewer XOR counts than optimal results reported before.
Highlights
In a block cipher, the linear diffusion layer is a significant component required for the security of the cipher
We focus on constructing maximum distance separable (MDS) matrices with as few XOR counts as possible
We extend some results about the involutory MDS matrix as follows: (1) We propose three theorems regarding the existences of involutory MDS matrices
Summary
The linear diffusion layer is a significant component required for the security of the cipher. The main idea is that selecting a matrix A that is sparse and compact in implementation, and calculate Ak to get an MDS matrix This method was successfully used for the constructions of hash function PHOTON [15], block cipher LED [16] and authenticated encryption scheme PRIMATEs [1]. Khoo et al [20] used the XOR count to measure the number of XORs required to compute the multiplication of a fixed element They showed that there are MDS matrices with higher Hamming weight than the AES diffusion matrix, but fewer XORs. After that, many works [23, 11, 25, 21, 22, 27, 28, 31] measured the lightweight of MDS matrices with XOR counts
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