Applicable Algebra in Engineering, Communication and Computing, Nov 23, 2020
Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is ... more Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that f (x)x −r is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie-Hellman mapping d(γ a) = γ a 2 , a = 0, 1,. .. , n − 1, and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(γ a , γ b) = γ ab , a, b = 0, 1,. .. , n − 1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.
Applicable Algebra in Engineering, Communication and Computing, 2020
Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is ... more Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that f (x)x −r is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie-Hellman mapping d(γ a) = γ a 2 , a = 0, 1,. .. , n − 1, and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(γ a , γ b) = γ ab , a, b = 0, 1,. .. , n − 1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.
Estimation of the minimum distance of cyclic codes is a classical problem in coding theory. Using... more Estimation of the minimum distance of cyclic codes is a classical problem in coding theory. Using the trace representation of cyclic codes and Hilbert's Theorem 90, Wolfmann found a general estimate for the minimum distance of cyclic codes in terms of the number of rational points on certain Artin-Schreier curves. In this thesis, we try to understand if Wolfmann's bound can be improved by modifying equations of the Artin-Schreier curves by the use of monomial and some nonmonomial permutation polynomials. Our experiments show that an improvement is possible in some cases.
Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinalit... more Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known. We consider a class $\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.
In this thesis we study several aspects of permutation polynomials over nite elds with odd charac... more In this thesis we study several aspects of permutation polynomials over nite elds with odd characteristic. We present methods of construction of families of complete mapping polynomials; an important subclass of permutations. Our work on value sets of non-permutation polynomials focus on the structure of the spectrum of a particular class of polynomials. Our main tool is a recent classi cation of permutation polynomials of Fq, based on their Carlitz rank. After introducing the notation and terminology we use, we give basic properties of permutation polynomials, complete mappings and value sets of polynomials in Chapter 1. We present our results on complete mappings in Fq[x] in Chapter 2. Our main result in Section 2.2 shows that when q > 2n + 1, there is no complete mapping polynomial of Carlitz rank n, whose poles are all in Fq. We note the similarity of this result to the well-known Chowla-Zassenhaus conjecture (1968), proven by Cohen (1990), which is on the non-existence of co...
We estimate the maximum-order complexity of a binary sequence in terms of its correlation measure... more We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order k cannot have very small maximum-order complexity.
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d ≥... more The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d ≥ 2 and any prime p > (d 2 − 3d + 4) 2 there is no complete mapping polynomial in F p [x] of degree d. For arbitrary finite fields F q , we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if n < ⌊q/2⌋, then there is no complete mapping in F q [x] of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank n < ⌊q/2⌋ are from being complete, by studying value sets of f + x. We provide examples of complete mappings if n = ⌊q/2⌋, which shows that the above bound cannot be improved in general.
Applicable Algebra in Engineering, Communication and Computing, Nov 23, 2020
Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is ... more Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that f (x)x −r is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie-Hellman mapping d(γ a) = γ a 2 , a = 0, 1,. .. , n − 1, and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(γ a , γ b) = γ ab , a, b = 0, 1,. .. , n − 1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.
Applicable Algebra in Engineering, Communication and Computing, 2020
Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is ... more Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that f (x)x −r is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie-Hellman mapping d(γ a) = γ a 2 , a = 0, 1,. .. , n − 1, and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(γ a , γ b) = γ ab , a, b = 0, 1,. .. , n − 1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.
Estimation of the minimum distance of cyclic codes is a classical problem in coding theory. Using... more Estimation of the minimum distance of cyclic codes is a classical problem in coding theory. Using the trace representation of cyclic codes and Hilbert's Theorem 90, Wolfmann found a general estimate for the minimum distance of cyclic codes in terms of the number of rational points on certain Artin-Schreier curves. In this thesis, we try to understand if Wolfmann's bound can be improved by modifying equations of the Artin-Schreier curves by the use of monomial and some nonmonomial permutation polynomials. Our experiments show that an improvement is possible in some cases.
Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinalit... more Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known. We consider a class $\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.
In this thesis we study several aspects of permutation polynomials over nite elds with odd charac... more In this thesis we study several aspects of permutation polynomials over nite elds with odd characteristic. We present methods of construction of families of complete mapping polynomials; an important subclass of permutations. Our work on value sets of non-permutation polynomials focus on the structure of the spectrum of a particular class of polynomials. Our main tool is a recent classi cation of permutation polynomials of Fq, based on their Carlitz rank. After introducing the notation and terminology we use, we give basic properties of permutation polynomials, complete mappings and value sets of polynomials in Chapter 1. We present our results on complete mappings in Fq[x] in Chapter 2. Our main result in Section 2.2 shows that when q > 2n + 1, there is no complete mapping polynomial of Carlitz rank n, whose poles are all in Fq. We note the similarity of this result to the well-known Chowla-Zassenhaus conjecture (1968), proven by Cohen (1990), which is on the non-existence of co...
We estimate the maximum-order complexity of a binary sequence in terms of its correlation measure... more We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order k cannot have very small maximum-order complexity.
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d ≥... more The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d ≥ 2 and any prime p > (d 2 − 3d + 4) 2 there is no complete mapping polynomial in F p [x] of degree d. For arbitrary finite fields F q , we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if n < ⌊q/2⌋, then there is no complete mapping in F q [x] of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank n < ⌊q/2⌋ are from being complete, by studying value sets of f + x. We provide examples of complete mappings if n = ⌊q/2⌋, which shows that the above bound cannot be improved in general.
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