Permutation polynomials over finite fields are an interesting and constantly active research subject of study for many years. They have important applications in areas of mathematics and engineering. In recent years, permutation binomials and permutation trinomials attract people's interests due to their simple algebraic forms. By reversely using Tu's method for the characterization of permutation polynomials with exponents of Niho type, we construct a class of permutation trinomials with coefficients 1 in this paper.As applications, two conjectures of [19] and a conjecture of [13] are all special cases of our result. To our knowledge, the construction method of permutation polynomials by polar decomposition in this paper is new. Moreover, we prove that in new class of permutation trinomials, there exists a permutation polynomial which is EA-inequivalent to known permutation polynomials for all m ≥ 2. Also we give the explicit compositional inverses of the new permutation trinomials for a special case.
Modular exponentiation and scalar multiplication are important operations in most public-key cryptosystems, and their efficient computation is essential to cryptosystems. The shortest addition chain is one of the most important mathematical concepts to realize the optimization of computation. However, finding a shortest addition chain of length r is generally regarded as an NP-hard problem, whose time complexity is comparable to O(r!). This paper proposes some novel methods to generate short addition chains. We firstly present a Simplified Power-tree method by deeply deleting the power-tree whose time complexity is reduced to O(r2). In this paper, a Cross Window method and its variant are introduced by improving the Window method. The Cross Window method uses the cross correlation to deal with the windows and its pre-computation is optimized by a new Addition Sequence Algorithm. The theoretical analysis is conducted to show the correctness and effectiveness. Meanwhile, our experiments show that the new methods can obtain shorter addition chains compared to the existing methods. The Cross Window method with the Addition Sequence algorithm can attain 44.74% and 9.51% reduction of the addition chain length, in the best case, compared to the Binary method and the Window method respectively.
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