D. Jungnickel, Finite Fields: Structure and Arithmetic, BI-Wissenschaftsverlag, Mannbeim, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....fields have received increasing interest in the literature. Finite fields are known to be very useful in finite geometries, combinatorics, algebraic coding theory, cryptology, combinatorial design theory, symbolic computations, pseudorandom number generation, and shift register sequences; cf. [32, 36, 28]. Let F q be a finite field with q elements and n the set of monic polynomials over F q of degree n. Assuming that each of the q polynomials in n is equally likely, we are interested in the random variable# n , counting the total number (i.e. counted with multiplicities) of irreducible ....

D. Jungnickel, Finite fields---structure and arithmetics, Bibliographisches Institut, Mannheim, 1993.

....fields have received increasing interest in the literature. Finite fields are known to be very useful in finite geometries, combinatorics, algebraic coding theory, cryptology, combinatorial design theory, symbolic computations, pseudorandom number generation, and shift register sequences; cf. [32, 36, 28]. Let F q be a finite field with q elements and P n the set of monic polynomials over F q of degree n. Assuming that each of the q polynomials in P n is equally likely, we are interested in the random This work was supported by National Science Council under the Grant NSC 85 2121 M 001 007. ....

D. Jungnickel, Finite fields---structure and arithmetics, Bibliographisches Institut, Mannheim, 1993.

....over GF (q) is given by I q (n) 1.1) where (d) is the M obius function. Less well known is the formula I 2 (n; 1) 1 2n d odd (d)2 ; 1. 2) which is the number of degree n irreducible polynomials over GF (2) with trace 1 (this can be inferred from results in Jungnickel [3], Section 2.7) One purpose of this paper is to re ne (1.1) and (1.2) by enumerating the irreducible degree n polynomials over GF (q) with a given trace. Carlitz [1] also solved this problem, arriving via a di erent technique at an expression that is di erent, but equivalent to the one given ....

....family of subsets of Z q , where Z q are the integers mod q. 3. Irreducible polynomials with given trace. In this section, the irreducible polynomials with a given trace are counted. We begin by introducing some notation that will be used in the remainder of the paper. We use Jungnickel [3] as a reference for terminology and basic results from nite eld theory. The trace of an element 2 GF (q ) over GF (q) is denoted T r( and is given by T r( 2 If 2 GF (q ) and d is the smallest positive integer for which = 1, then f(x) is the ....

Dieter Jungnickel, Finite Fields: structure and arithmetics, B.I. Wissenschaftsverlag, 1993.

.... 2 Finite Fields, Their Characters and the Dual BCH Codes We set out some facts concerning the trace map on a finite field, assuming the reader to be familiar with the basic properties of finite fields (existence, uniqueness, primitive elements and so on) Basic references for finite fields are [23, 31, 32]. We will almost exclusively be concerned with fields of characteristic two in this paper, though almost everything we say can be generalised to characteristic p with appropriate modifications. Throughout, m;n will denote positive integers with mjn. Also, F 2 n denotes the finite field with 2 ....

.... j (ff ) ji ; 0 i 2 Gamma 1: The maps j are called the multiplicative characters of F 2 n : they are homomorphisms from (F 2 n ; Delta) to U . The map 0 is called the trivial multiplicative character. For much more information about characters of finite fields, see [22, 23, 32] Next we define the main class of codes that we ll work with in this paper, the dual BCH codes. In fact, we work with a sub class of these codes, more properly called binary, primitive, dual BCH codes. Let ff be primitive in F 2 n and let t be a positive integer with 1 2t Gamma 1 2 1. ....

D. Jungnickel. Finite Fields --- Structure and Arithmetics. B.I. Wissenschaftsverlag, Mannheim, 1993.

....family of subsets of Z q , where Z q are the integers mod q. 3. Irreducible polynomials with given trace. In this section, the irreducible polynomials with a given trace are counted. We begin by introducing some notation that will be used in the remainder of the paper. We use Jungnickel [3] as a reference for terminology and basic results from finite field theory. The trace of an element # # GF (q n ) over GF (q) is denoted T r(#) and is given by T r(#) # # q # q 2 # q n 1 . If # # GF (q n ) and d is the smallest positive integer for which # q d = 1, ....

D. Jungnickel, Finite Fields: Structure and arithmetics, B.I. Wissenschaftsverlag, Mannheim, Germany, 1993.

....and evaluating their security. For a summary of this work, consult [18] A description of a hardware implementation of an elliptic curve cryptosystem can be found in [2] Three good references on the theory of finite fields are the books by McEliece [17] Lidl and Niederreiter [16] and Jungnickel [10]. The article [1] discusses how to efficiently perform arithmetic operations in finite fields of characteristic 2. A hardware implementation of arithmetic in such fields which exploits the properties of so called optimal normal bases is described in [3] The ElGamal public key encryption and ....

D. Jungnickel, Finite Fields: Structure and Arithmetics, B.I.-Wissenschaftsverlag, Mannheim, 1993.

....technical results which are then used to count the elements in GF (2 n ) with given trace and subtrace. In Section 5, we prove the main theorem. Congruences modulo 4 are crucial and pervasive in this paper and expressions of the form x j y (mod 4) are shortened to read x j y. We use Jungnickel [6] as a reference for terminology and basic results from finite field theory. 2 A generalized Mobius inversion formula The approach we follow in this section is similar to that found in Knuth, Graham, and Patashnik [7] The defining property of the Mobius function is X djn (d) n = 1] 4) ....

.... St(fi) is the coefficient of x n Gamma2 in f(x) n Gamma1 Y i=0 (x fi 2 i ) We also write St(f) instead of St(fi) For polynomial f of degree n over GF(2) define f Q (x) x n f(x x Gamma1 ) The following three results are all stated and proven in the book of Jungnickel [6]. Lemma 2 ( 6] pg. 77) Let g be any monic self reciprocal polynomial of degree 2n over GF(2) Then there exists a polynomial f of degree n over GF(2) such that g = f Q . If g is irreducible, then f is also irreducible. Theorem 2 ( 6] pg. 77) The number of monic self reciprocal irreducible ....

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Dieter Jungnickel, Finite Fields: structure and arithmetics, B.I. Wissenschaftsverlag, 1993.

....to F q n , and the set of Gauss periods of type (n; k) does not depend on the particular choice of fi as a primitive rth root of unity. Gauss periods have been used to construct normal bases in finite fields by Mullin et al. 1989) and Ash et al. 1989) The texts by Menezes et al. 1993) and Jungnickel (1993) present detailed discussions. Wassermann (1990) gives the exact condition for a Gauss period of type (n; k) to form a normal basis for F q n over F q . We give a form of Wasserman s condition that is somewhat easier to handle computationally. Theorem 3.1. Let r = nk 1 be a prime not dividing ....

....et al. 1989) and certain Gauss periods. In particular, when k = 1, or k = 2 and q = 2, the normal bases generated by Gauss periods are optimal. See also Menezes et al. 1993, Chapter 5) Since self dual bases are useful in implementing finite fields (Berlekamp 1982, Geiselmann and Gollmann 1989, Jungnickel 1993, Wang 1989) we next determine when normal bases formed by Gauss periods are self dual. Let (fl 1 ; fl 2 ; fl n ) and (ffi 1 ; ffi 2 ; ffi n ) be two bases for F q n over F q . They are said to be dual to each other if T (fl i ffi j ) is 0 when i 6= j, and 1 when i = j, where T ....

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Jungnickel, D. (1993). Finite Fields: Structure and Arithmetics. Wissenschaftsverlag, Mannhein-LeipzigWien -Zurich.

.... k i a i b k Gammai : 4.5) To prove this, we show that the elements of any nonzero LRS of a given polynomial can be written in terms of the trace function (this was stated in [10] but not proven) and then use the fact that the degrees of f and g are coprime to obtain (4.5) Lemma 4. 5 ([5]) Let A = fff 0 ; ff 1 ; ff m Gamma1 g and B = ffl 0 ; fl 1 ; fl m Gamma1 g be a pair of dual bases of F q m=F q , and let be any element of F q m . Then the coordinate x i of ff i in = x 0 ff 0 x 1 ff 1 Delta Delta Delta xm Gamma1 ff m Gamma1 equals T r( fl i ) and ....

....can be made for the irreducible g of degree n over F q mentioned above. For this polynomial, we use the notation fi for its roots, fb k g for its associated nonzero LRS, and let ae 2 F q n play the same role as above. We require one more result before proving Theorem 4.4. Lemma 4. 7 ([5]) Let A = fff 0 ; ff 1 ; ff m Gamma1 g and B = ffi 0 ; fi 1 ; fi n Gamma1 g be bases for K = F q m and L = F q n over F = F q , respectively, and assume that m and n are coprime. Then one has the following results, where we write E = F q mn . 14 JOEL V. BRAWLEY, SHUHONG GAO, AND ....

Jungnickel, Dieter. Finite Fields: Structure and Arithmetics. Wissenschaftsverlag, Mannheim, 1993.

....and evaluating their security. For a summary of this work, consult [16] A description of a hardware implementation of an elliptic curve cryptosystem can be found in [2] Three good references on the theory of finite fields are the books by McEliece [15] Lidl and Niederreiter [14] and Jungnickel [10]. The article [1] discusses how to efficiently perform arithmetic operations in finite fields of characteristic 2. A hardware implementation of arithmetic in such fields which exploits the properties of so called optimal normal bases is described in [3] The ElGamal public key encryption and ....

D. Jungnickel, Finite Fields: Structure and Arithmetics, B.I.-Wissenschaftsverlag, Mannheim, 1993.

No context found.

D. Jungnickel, Finite Fields: Structure and Arithmetic, BI-Wissenschaftsverlag, Mannbeim, 1993.

No context found.

D. Jungnickel, Finite Fields: Structure and arithmetics, B.I. Wissenschaftsverlag, Mannheim, Germany, 1993.

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D. Jungnickel, Finite Fields: Structure and Arithmetics, Bibliographisches Institut, Mannheim, 1993.

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D. Jungnickel, Finite Fields: Structure and Arithmetics, Bibliographisches Institut, Mannheim, 1993.

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D. Jungnickel, Finite Fields: Structure and Arithmetics, Mannheim: Wissenschaftsverlag (1992).

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