A formula on the complexity of the normal bases generated by prime Gauss period overfinite fields is presented in terms of cyclotomic numbers.Then,the authors determine explicitly thecomplexity of such normal bases an...A formula on the complexity of the normal bases generated by prime Gauss period overfinite fields is presented in terms of cyclotomic numbers.Then,the authors determine explicitly thecomplexity of such normal bases and their dual bases in several cases where the related cyclotomicnumbers have been calculated.Particularly,the authors find several series of such normal bases withlow complexity.展开更多
In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index?n in xi. We will only determine 1003 for an index n odd and a degree . The others wil...In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index?n in xi. We will only determine 1003 for an index n odd and a degree . The others will be deduced, we’ll see how. The formula is , without exception if u(n)=-1?or if 4 doesn’t divide and with its 165 exceptions of which 7 when u(n)=0?and 158 when u(n)=1?that will be shared in 154 and 4 pairs (n, i), which we will specify the conditions and values of the coefficients. According to u(n), according to the class of i modulo p, the first factor of the prime factor decomposition of n when u(n)=1?and according to gcd(n, i), the formula will or will not be valid and replaced otherwise by the good value that will be 0 for 152 pairs (n,i) or 1 in the 13 other exceptions.展开更多
In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the correspond...In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.展开更多
基金supported by the National Fundamental Science Research Program 973 of China under Grant No. 2004 CB3180000the State Key Lab. (Information Security) of China
文摘A formula on the complexity of the normal bases generated by prime Gauss period overfinite fields is presented in terms of cyclotomic numbers.Then,the authors determine explicitly thecomplexity of such normal bases and their dual bases in several cases where the related cyclotomicnumbers have been calculated.Particularly,the authors find several series of such normal bases withlow complexity.
文摘In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index?n in xi. We will only determine 1003 for an index n odd and a degree . The others will be deduced, we’ll see how. The formula is , without exception if u(n)=-1?or if 4 doesn’t divide and with its 165 exceptions of which 7 when u(n)=0?and 158 when u(n)=1?that will be shared in 154 and 4 pairs (n, i), which we will specify the conditions and values of the coefficients. According to u(n), according to the class of i modulo p, the first factor of the prime factor decomposition of n when u(n)=1?and according to gcd(n, i), the formula will or will not be valid and replaced otherwise by the good value that will be 0 for 152 pairs (n,i) or 1 in the 13 other exceptions.
基金supported by National Natural Science Foundation of China(Grant Nos.10990011,11001145 and 61170289)the Science and Technology on Information Assurance Laboratory Foundation(Grant No.KJ-12-01)
文摘In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.
