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Enumerating Permutation Polynomials over Finite Fields by Degree
, 2002
"... f s 5q # 2g: Then, TfiS confirms the common belief that almost all permutation polynomials have degree q # 2. T. first few values of N are listed below: q 23457 8 9 11 N 0 0 12 20 630 5368 42 120 3 634 950 q # 1! 1 2 6 24 720 5040 40 320 3 628 800 Proof.To proof uses exponential sums an ..."
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Cited by 6 (0 self)
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f s 5q # 2g: Then, TfiS confirms the common belief that almost all permutation polynomials have degree q # 2. T. first few values of N are listed below: q 23457 8 9 11 N 0 0 12 20 630 5368 42 120 3 634 950 q # 1! 1 2 6 24 720 5040 40 320 3 628 800 Proof.To proof uses exponential sums
Enumerating Permutation Polynomials I: Permutations with NonMaximal Degree
, 2002
"... s can be found in the book of Lidl and Niederreiter [5]. Various applications of permutation polynomials to cryptography have been described. See for example [1,2]. Lidl and Mullen in [3,4] (see also [6]) describe a number of open problems regarding permutations polynomials: among these, the problem ..."
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Cited by 3 (2 self)
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, the problem of enumerating permutation polynomials by their degree. We denote by S s fx 2 F sx=xg the set of those elements of F q which are moved by s. Our first remark is that @f s #q#j if s=id: 2 T see this it is enough to note that the polynomial f s x#x has as roots all the elements of F q fixed
Corrigendum Corrigendum to “enumerating permutation polynomials—I: Permutations with nonmaximal degree”
, 2005
"... Let Fq be a finite field with q elements and suppose C is a conjugation class of permutations of the elements of Fq. We denote by C = (c1; c2;...; ct) the conjugation class of permutations that admit a cycle decomposition with ci icycles (i = 1,...,t). Further, we set c = 2c2 +···+tct = q − c1 to b ..."
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to be the number of elements of Fq moved by any permutation in C. If � ∈ C, then the permutation polynomial associated to � is defined as f�(t) = � Therefore for q>3 the function x∈Fq �(x) 1 − (t − x) q−1�
Corrigendum Corrigendum to “enumerating permutation polynomials—I: Permutations with nonmaximal degree”
, 2005
"... Let Fq be a finite field with q elements and suppose C is a conjugation class of permutations of the elements of Fq. We denote by C = (c1; c2;...; ct) the conjugation class of permutations that admit a cycle decomposition with ci icycles (i = 1,..., t). Further, we set c = 2c2 + · · ·+ tct = q−c1 ..."
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to be the number of elements of Fq moved by any permutation in C. If! ∈ C, then the permutation polynomial associated to! is defined as f!(t) = x∈Fq
Enumerating Permutation Polynomials II: kcycles with minimal degree
, 2003
"... We consider the function m [k] (q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upperbound (k1)!(q(q1))/k for char(Fq ) > e and the lowerbound #(k)(q(q1))/ ..."
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Cited by 2 (2 self)
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We consider the function m [k] (q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upperbound (k1)!(q(q1))/k for char(Fq ) > e and the lowerbound #(k)(q(q1
Enumerating permutation polynomials II: kcycles with minimal degree
"... We consider the function mkðqÞ that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upperbound mkðqÞpðk 1Þ!ðqðq 1ÞÞ=k for charðFqÞ4eðk3Þ=e and the lowerbound mkðqÞXjðkÞðqðq 1ÞÞ=k ..."
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We consider the function mkðqÞ that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upperbound mkðqÞpðk 1Þ!ðqðq 1ÞÞ=k for charðFqÞ4eðk3Þ=e and the lowerbound mkðqÞXjðkÞðqðq
A Digital Signature Scheme Secure Against Adaptive ChosenMessage Attacks
, 1995
"... We present a digital signature scheme based on the computational diculty of integer factorization. The scheme possesses the novel property of being robust against an adaptive chosenmessage attack: an adversary who receives signatures for messages of his choice (where each message may be chosen in a ..."
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Cited by 985 (43 self)
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were considered in the folklore to be contradictory. More generally, we show how to construct a signature scheme with such properties based on the existence of a "clawfree" pair of permutations  a potentially weaker assumption than the intractibility of integer factorization. The new scheme
Algorithms for Quantum Computation: Discrete Logarithms and Factoring
, 1994
"... A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a increase in computation time of at most a polynomial factor. It is not clear whether this is still true when quantum mechanics is taken into consi ..."
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Cited by 1103 (7 self)
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A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a increase in computation time of at most a polynomial factor. It is not clear whether this is still true when quantum mechanics is taken
Simulating Physics with Computers
 SIAM Journal on Computing
, 1982
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 601 (1 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration
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