Abstract
Costas arrays special permutation matrices have widespread applications in many fields such as radar signal design and cryptography. However, so far the two basic problems-the existence and counting problems remain unsolved. This paper discusses the number of ntimesn symmetric Costas arrays, and discloses the relationship between the number of Costas arrays and symmetric ones. Also we attempt to search for Costas arrays based on simulated annealing algorithm (SAA). System algebraic methods can construct lots of Costas arrays but not all. For a long time, people have just enumerated all costas arrays for orders less than 26 with exponential computational complexity. On the other hand, it's easy to check whether a permutation matrix is a Costas array or not with polynomial computational complexity. Based on the point, we present a signature scheme in the paper.
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