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S-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for ( n  + 1)-bit S-boxes in [ 12 ]. We obtain the nonlinearity of ( n  + 1)-bit S-boxes, and one necessary and sufficient conditions of ( n  + 1)-bit S-boxes satisfying  m -order resilient. Meanwhile, we also give one characterization of ( n  + 1)-bit S-boxes satisfying  t -order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an  n -bit S-box  S  0  and the ( n  + 1)-bit S-box  S  (which is constructed by  S  0 ).

On cryptographic properties of ( n + 1)-bit S-boxes constructed by known n -bit S-boxes