Open AccessOn Efficient Constructions of Lightweight MDS Matrices
Author(s) -
Lijing Zhou,
Licheng Wang,
Yiru Sun
Publication year - 2018
Publication title -
iacr transaction on symmetric cryptology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.715
H-Index - 10
ISSN - 2519-173X
DOI - 10.46586/tosc.v2018.i1.180-200
Subject(s) - hadamard transform , mathematics , hadamard matrix , complex hadamard matrix , involutory matrix , hadamard product , matrix (chemical analysis) , separable space , binary number , bitwise operation , discrete mathematics , arithmetic , algebra over a field , pure mathematics , computer science , square matrix , symmetric matrix , mathematical analysis , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , programming language
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.