In this paper, we provide several methods to construct nonlinear resilient functions with multiple good cryptographic properties, including high nonlinearity, high algebraic degree, and non-existence of linear structures. Firstly, we present an improvement on a known construction of resilient S-boxes such that the nonlinearity and the algebraic degree will become higher in some cases. Then a construction of highly nonlinear t-resilient Boolean functions without linear structures is given, whose algebraic degree achieves n-t-1, which is optimal for n-variable t-resilient Boolean functions. Furthermore, we construct a class of resilient S-boxes without linear structures, which possesses the highest nonlinearity and algebraic degree among all currently known constructions.

Aggregate signature (AS) schemes enable anyone to compress signatures under different keys into one. In sequential aggregate signature (SAS) schemes, the aggregate signature is computed incrementally by the sighers. Several trapdoor-permutation-based SAS have been proposed. In this paper, we give a constructions of SAS based on the first SAS scheme with lazy verification proposed by Brogle et al. in ASIACRYPT 2012. In Brogle et al.'s scheme, the size of the aggregate signature is linear of the number of the signers. In our scheme, the aggregate signature has constant length which satisfies the original ideal of compressing the size of signatures.