TAILIEUCHUNG - Báo cáo toán học: " Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions. | Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions Francis N. Castro Department of Mathematics University of Puerto Rico San Juan PR 00931 Luis A. Medina Department of Mathematics University of Puerto Rico San Juan PR 00931 Submitted Jan 28 2011 Accepted May 13 2011 Published May 25 2011 Mathematics Subject Classification 11T23 05E05 Dedicated to Doron Zeilberger on the occasion of his 60th birthday Abstract In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular when the degree of the symmetric function is a power of two then the exponential sum is much smaller than 2 . Keywords Exponential sums recurrences Cusick et al. Conjecture for elementary balanced symmetric boolean functions 1 Introduction Boolean functions are one of the most studied objects in mathematics. They are important in many applications for example in the design of stream ciphers block and hash functions. These functions also play a vital role in cryptography as they are used as filter and combination generator of stream ciphers based on linear feed-back shift registers. The THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P8 1 case of boolean functions of degree 2 has been intensively studied because of its relation to bent functions see 11 1 . One can find many papers and books discussing the properties of boolean functions see 5 9 2 and 6 . The subject can be studied from the point of view of complexity theory or from the algebraic point of view as we do in this paper where we compute the asymptotic behavior of exponential sums of symmetric boolean functions. The .

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