On the construction of \(20 \times 20\) and \(2 4 \times 24\) binary matrices with good implementation properties for lightweight block ciphers and hash functions (Q1718633)

Summary: We present an algebraic construction based on state transform matrix (companion matrix) for \(n \times n\) (where \(n \neq 2^k\), \(k\) being a positive integer) binary matrices with high branch number and low number of fixed points. We also provide examples for \(20 \times 20\) and \(24 \times 24\) binary matrices having advantages on implementation issues in lightweight block ciphers and hash functions. The powers of the companion matrix for an irreducible polynomial over \(\text{GF}(2)\) with degree 5 and 4 are used in finite field Hadamard or circulant manner to construct \(20 \times 20\) and \(24 \times 24\) binary matrices, respectively. Moreover, the binary matrices are constructed to have good software and hardware implementation properties. To the best of our knowledge, this is the first study for \(n \times n\) (where \(n \neq 2^k\), \(k\) being a positive integer) binary matrices with high branch number and low number of fixed points.