Previously proposed batch signature schemes do not allow a signer to generate a signature immediately for sequentially asked signing queries. In this letter, we propose flexible batch signatures which do not need any waiting period and have very light computational overhead. Therefore our schemes are well suited for low power devices.

The differential uniformity, the boomerang uniformity, and the extended Walsh spectrum etc are important parameters to evaluate the security of S (substitution)-box. In this paper, we introduce efficient formulas to compute these cryptographic parameters of permutation polynomials of the form xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 is even, where d=3 is the smallest nontrivial d for even n. We also investigate the differential uniformity of some permutation polynomials introduced in some recent papers for the case d=2n/2+1.

We propose a new linear array for multiplication in GF(2m) which outperforms most of the existing linear multipliers in terms of the area and time complexity. Moreover we will give a very detailed comparison of our array with other existing architectures for the five binary fields GF(2m), m=163,233,283,409,571, recommended by NIST for elliptic curve cryptography.