## Normal Bases over Finite Fields (1993)

Citations: | 10 - 0 self |

### BibTeX

@MISC{Gao93normalbases,

author = {Shuhong Gao},

title = {Normal Bases over Finite Fields},

year = {1993}

}

### OpenURL

### Abstract

Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...

### Citations

2740 | New Directions in Cryptography
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- 1976
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Citation Context ...onentiation using the repeated square and multiply method can be speeded up, especially if q = 2. This is very important in the implementation of such cryptosystems as the Diffie-Hellman key exchange =-=[42]-=- and ElGamal cryptosystem [44] where one needs to compute large powers of elements in a fixed finite field. Let the t (k) ij terms be defined by (1.4). Raising both sides of equation (1.4) to the q−ℓ ... |

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- 1974
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Citation Context ... square and multiply method can be speeded up, especially if q = 2. This is very important in the implementation of such cryptosystems as the Diffie-Hellman key exchange [42] and ElGamal cryptosystem =-=[44]-=- where one needs to compute large powers of elements in a fixed finite field. Let the t (k) ij terms be defined by (1.4). Raising both sides of equation (1.4) to the q−ℓ -th power, one finds that t (ℓ... |

540 |
A classical introduction to modern number theory. Second edition. Graduate Texts
- Ireland, Rosen
- 1990
(Show Context)
Citation Context ...primes. Then (a) (−1/p) =(−1) (p−1)/2 , (b) (2/p) =(−1) (p2 −1)/8 , (c) (p/q) =(−1) ((p−1)/2)((q−1)/2) (q/p). Proof: It is our purpose to give a proof of (c), the proof of (a) and (b) can be found in =-=[68]-=-. Note that q|(pq−1 − 1). There is a primitive q-th root of unity in Fpq−1, sayξ.Asξ�=1,ξmust be a root of (xq − 1)/(x − 1), that is, �q−1 ξ i =0. i=0 Now let S be the set of quadratic residues in Fq ... |

527 | Finite Fields - Lidl, Niederreiter - 1983 |

480 |
Basic Algebra I
- Jacobson
- 1974
(Show Context)
Citation Context ...ain why we are interested in normal bases and give a brief overview of the thesis. We will assume that one is familiar with the basic concepts for field extensions; our standard reference is Jacobson =-=[69]-=-. Let F be field and E be a finite Galois extension of F of degree n and Galois group G. A normal basis of E over F is a basis of the form {σα : σ ∈ G} where α ∈ E. That is, a normal basis consists of... |

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430 |
zur Gathen and
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410 |
Introduction to Finite Fields and Their Applications
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- 1986
(Show Context)
Citation Context .... ⎜ . ⎝ . Tr(α1α2) Tr(α2α2) . ··· ··· Tr(α1αn) ⎟ Tr(α2αn) ⎟ . ⎟, . ⎟ ⎠ Tr(αnα1) Tr(αnα2) ··· Tr(αnαn) where Tr is understood to be Tr q n |q. Obviously, ∆(α1,... ,αn)∈Fq. Theorem 2.2.1 (Theorem 2.37, =-=[89]-=-) For any n elements α1,... ,αnin Fqn, they form a basis of Fqn over Fq if and only if ∆(α1,... ,αn)�=0. Proof: First assume that α1,... ,αnform a basis for Fqn over Fq. We prove that ∆(α1,... ,αn)�= ... |

168 |
Introduction to Cyclotomic Fields
- Washington
- 1982
(Show Context)
Citation Context ...) of the theorem are satisfied. All assertions of (ii) have been proved. � 4.3 Constructing Irreducible Polynomials under ERH The α in Theorem 4.1.4 has classical origins and is called a Gauss period =-=[147, 110]-=-. Gauss periods are used to realize the Galois correspondence between subfields of a cyclotomic field and subgroups of its Galois group, as shown in section 1.1. Gauss periods are also useful for inte... |

111 |
Finite fields for computer scientists and engineers
- McEliece
- 1987
(Show Context)
Citation Context ...nsCHAPTER 1. INTRODUCTION 6 algorithm to make a hardware or software design of a finite field feasible for large n. One example is the bit-serial multiplication scheme due to Berlekamp [16], see also =-=[97]-=-, and its generalizations [101, 55, 57, 143, 63, 134] using a pair of (dual) bases. In the following we examine the MasseyOmura scheme [95] which exploits the symmetry of normal bases. A normal basis ... |

100 |
Zassenhaus: Algorithmic algebraic number theory, revised reprint of the 1989 original
- Pohst, H
- 1997
(Show Context)
Citation Context ...) of the theorem are satisfied. All assertions of (ii) have been proved. � 4.3 Constructing Irreducible Polynomials under ERH The α in Theorem 4.1.4 has classical origins and is called a Gauss period =-=[147, 110]-=-. Gauss periods are used to realize the Galois correspondence between subfields of a cyclotomic field and subgroups of its Galois group, as shown in section 1.1. Gauss periods are also useful for inte... |

98 |
An implementation of elliptic curve cryptosystems over F2155
- Agnew, Mullin, et al.
- 1993
(Show Context)
Citation Context ...ed. Work in this area has resulted in several hardware and software designs or implementations [41, 45, 121, 145, 146, 151], including single-chip exponentiators for the fields F 2 127 [152], F 2 155 =-=[3]-=-, and F 2 332 [56], and an encryption processor for F 2 593 [114] for public key cryptography. These products are based on multiplication schemes due to Massey and Omura [95] and Mullin, Onyszchuk and... |

79 |
Fundamentals of Number Theory
- LeVeque
- 1977
(Show Context)
Citation Context ...). So one can compute a 2eth primitive root of unity in Fp2 quickly for any given integer e and prime p of the form 2Nk −1. Also, for fixed N =2e , the prime number theorem in arithmetic progressions =-=[88]-=- implies that the number of primes 2Nk −1 ≤ N2 is approximately N/(2e log 2). This means that primes of the required form exist in reasonable abundance and their sizes can be bounded by N 2 . So the p... |

68 | New algorithms for finding irreducible polynomials over finite fields - Shoup - 1990 |

53 |
Applications of Finite Fields
- Menezes, Blake, et al.
- 1993
(Show Context)
Citation Context ...lynomial over Fq, and let b ∈ Fq. Letpbethe characteristic of Fq. Then the polynomial P (xp − x − b) is irreducible over Fq if and only if Trq|p(nb − cn−1) �= 0. A proof of this lemma can be found in =-=[89, 99]-=-. The next theorem is due to Varshamov [139], where no proof is given. Theorem 3.4.10 (Varshamov) Let p be a prime and let f(x) =xn + �n−1 i=0 cixi be irreducible over Fp. Suppose that there exists an... |

51 |
Computational method and apparatus for finite field arithmetic
- Omura, Massey
- 1986
(Show Context)
Citation Context ...lds F 2 127 [152], F 2 155 [3], and F 2 332 [56], and an encryption processor for F 2 593 [114] for public key cryptography. These products are based on multiplication schemes due to Massey and Omura =-=[95]-=- and Mullin, Onyszchuk and Vanstone [105] by using normal bases to represent finite fields and choosing appropriate algorithms for the arithmetic. Interestingly, the advantage of using a normal basis ... |

49 |
An implementation for a fast public-key cryptosystem
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- 1991
(Show Context)
Citation Context ...nections between registers containing the elements A, B and C. The fanout of a cell is the number of connections to the cell, and should be as small as possible. Agnew, Mullin, Onyszchuk and Vanstone =-=[2]-=- designed a different architecture with a low fanout, and they successfully implemented the field F2593 in hardware (see [114]). Since this scheme is more complicated, we omit its description here. We... |

39 | Searching for primitive roots in finite fields
- Shoup
- 1992
(Show Context)
Citation Context ...the case that q is a prime and by Lenstra and Schoof [87] in 1987 for the general case. For the construction of primitive normal bases and primitive elements, see Cohen [34], Hachenberger [60], Shoup =-=[129]-=-, Stepanov and Shparlinskiy [131, 132, 133]. An important class of normal bases are self-dual normal bases. More generally, one has the concept of self-dual bases, which is useful for construction of ... |

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33 |
Cyclotomy and Difference Sets
- Storer
- 1967
(Show Context)
Citation Context ... − 1 (except for i = i0 where 1+τv0qi0≡0(modnk + 1)). These numbers are called cyclotomic numbers in the theory of cyclotomy and their values are determined for many small values of n, for detail see =-=[135]-=-. We remark that αi = αqi = �k−1 v=0 βτ v q i does not depends on the particular τ and q, it depends only on the coset {q i ,q i τ,... ,q i τ k−1 }(modm = kn + 1) of the unique subgroup <τ>of order k ... |

30 |
Finding irreducible polynomials over finite fields
- Adleman, Lenstra
- 1986
(Show Context)
Citation Context ...xity of normal bases and for optimal normal bases is given in the next section. We mention that Gauss periods are also useful in integer factorization [12] and construction of irreducible polynomials =-=[1]-=- (refer to section 4.3). 1.2 Finite Field Arithmetic Let q be a prime power and n a positive integer. Let Fq and Fqn be finite fields of q and qn elements, respectively. Let us first look at how addit... |

28 |
Bit-serial Reed-Solomon encoder
- Berlekamp
- 1982
(Show Context)
Citation Context ...e multiplicationsCHAPTER 1. INTRODUCTION 6 algorithm to make a hardware or software design of a finite field feasible for large n. One example is the bit-serial multiplication scheme due to Berlekamp =-=[16]-=-, see also [97], and its generalizations [101, 55, 57, 143, 63, 134] using a pair of (dual) bases. In the following we examine the MasseyOmura scheme [95] which exploits the symmetry of normal bases. ... |

28 | On arithmetical algorithms over finite fields - Cantor - 1989 |

28 | VLSI Designs for Multiplications over Finite Fields GF(2 m - Mastrovito - 1988 |

24 |
Low complexity normal bases
- Ash, Blake, et al.
- 1989
(Show Context)
Citation Context ...e of both type I and type II optimal normal bases, otherwise there exists only a type II optimal normal basis. The constructions in Theorems 4.1.1 and 4.1.2 are generalized by Ash, Blake and Vanstone =-=[10]-=- and further by Wassermann [148] to construct normal bases of low complexity as in Theorem 4.1.4. To establish this result, we first prove a lemma. Lemma 4.1.3 Let k, n be integers such that nk +1 is ... |

23 | Optimal normal bases
- Gao, Lenstra
- 1992
(Show Context)
Citation Context ...s must be either of type I or type II. Later Gao [49] proved that any optimal normal basis of a finite field must be equivalent to a type I or a type II optimal normal basis. Finally, Gao and Lenstra =-=[50]-=- extended the result to any finite Galois extension of an arbitrary field. In this section we prove that all the optimal normal bases in finite fields are completelysCHAPTER 4. OPTIMAL NORMAL BASES 74... |

23 |
Optimal normal bases
- Mullin, Onyszchuk, et al.
- 1988
(Show Context)
Citation Context ... prove that j=0 t (k) ij = ti−j,k−j, for all i, j, k. Therefore the number of non-zero entries in T0 is equal to the number of non-zero entries in T . Following Mullin, Onyszchuk, Vanstone and Wilson =-=[103]-=-, we call the number of non-zero entriessCHAPTER 1. INTRODUCTION 7 in T the complexity of the normal basis N, denoted by cN . Since the matrices {Tk} are uniquely determined by T , we call T the multi... |

22 |
Forms in odd degree extensions and self-dual normal bases
- Bayer–Fluckiger, Lenstra
- 1990
(Show Context)
Citation Context ...dual normal basis over Fq if and only if both n and q are odd or q is even and n is not divisible by 4. σ∈GsCHAPTER 1. INTRODUCTION 11 The above two theorems are proved by Bayer-Fluckiger and Lenstra =-=[13, 14]-=-. Partial results were obtained earlier by Lempel, and Weinberger [79, 80, 82], Imamura and Morii [67, 100], Beth, Fummy and Mühlfeld [19], Kersten and Michaliček [75], Conner and Perlis [37]. For enu... |

21 | On the deterministic complexity of factoring polynomials over finite fields - Shoup - 1990 |

21 |
I.: VLSI architectures for computing multiplications and inverses in GF (2 m
- Wang, Troung, et al.
- 1985
(Show Context)
Citation Context ...g finite fields, the implementation of finite field arithmetic, in either hardware or software, is required. Work in this area has resulted in several hardware and software designs or implementations =-=[41, 45, 121, 145, 146, 151]-=-, including single-chip exponentiators for the fields F 2 127 [152], F 2 155 [3], and F 2 332 [56], and an encryption processor for F 2 593 [114] for public key cryptography. These products are based ... |

20 |
Factoring with cyclotomic polynomials
- Bach, Shallit
- 1989
(Show Context)
Citation Context ...timal normal bases. The definition for the complexity of normal bases and for optimal normal bases is given in the next section. We mention that Gauss periods are also useful in integer factorization =-=[12]-=- and construction of irreducible polynomials [1] (refer to section 4.3). 1.2 Finite Field Arithmetic Let q be a prime power and n a positive integer. Let Fq and Fqn be finite fields of q and qn elemen... |

19 |
Factor refinement
- Bach, Driscoll, et al.
- 1993
(Show Context)
Citation Context ....1 tells us that if q>2n(n−1), then θ is a normal element with probability at least 1/2. The entire computation takes O((n + log q)(n log q) 2 ) bit operations, as shown by Bach, Driscoll and Shallit =-=[11]-=-. Frandsen [46] shows that when q>2n(n−1), an arbitrary element in Fqn is a normal element with probability ≥ 1/2. In general, he proves that a random element in Fqn is a normal element with probabili... |

18 |
A Comparison of VLSI Architecture of Finite Field Multipliers Using Dual
- Hsu, Truong, et al.
- 1988
(Show Context)
Citation Context ...gorithm to make a hardware or software design of a finite field feasible for large n. One example is the bit-serial multiplication scheme due to Berlekamp [16], see also [97], and its generalizations =-=[101, 55, 57, 143, 63, 134]-=- using a pair of (dual) bases. In the following we examine the MasseyOmura scheme [95] which exploits the symmetry of normal bases. A normal basis of Fqn over Fq is a basis of the form {α, αq ,... ,αq... |

18 |
Finding isomorphisms between finite fields
- Lenstra
- 1991
(Show Context)
Citation Context ...rrently no deterministic polynomial time algorithm known to factor xn−1 when p is large. In the following we will present two deterministic polynomial time algorithms due to Lüneburg [92] and Lenstra =-=[85]-=-. As shown by Bach, Driscoll and Shallit [11], both algorithms have the same complexity. In both algorithms we need to find the σ-Order Ordθ(x) of an arbitrary element θ in Fqn. Note that the degree o... |

16 |
Contributions to the theory of finite fields
- Ore
- 1934
(Show Context)
Citation Context ...s a normal basis of Fqn over Fq. As another consequence of Theorem 2.4.5, we count the number of normal elements, and thus the number of normal bases of Fqn over Fq. Corollary 2.4.7 (Hensel [61], Ore =-=[106]-=-) The total number of normal elements in Fqn over Fq is v(n, q) = r� q di(t−1) (q di − 1), and the number of normal bases of Fq n over Fq is v(n, q)/n. i=1 Proof: The first statement is obvious from T... |

16 | A VLSI architecture for fast inversion in GF(2 m - Feng - 1989 |

15 |
Distribution of primitive roots in a finite field
- Carlitz
- 1953
(Show Context)
Citation Context ...imitive normal basis of Fqn over Fq we mean a normal basis {α, αq ,... ,αqn−1} such that α also generates the multiplicative group of Fqn. This result was proved by CarlitzsCHAPTER 1. INTRODUCTION 10 =-=[32]-=- in 1952 for qn sufficiently large, by Davenport [38] in 1968 for the case that q is a prime and by Lenstra and Schoof [87] in 1987 for the general case. For the construction of primitive normal bases... |

14 |
A survey of trace forms of algebraic number fields
- Conner
- 1984
(Show Context)
Citation Context ...stra [13, 14]. Partial results were obtained earlier by Lempel, and Weinberger [79, 80, 82], Imamura and Morii [67, 100], Beth, Fummy and Mühlfeld [19], Kersten and Michaliček [75], Conner and Perlis =-=[37]-=-. For enumeration of self-dual normal bases, see Lempel and Seroussi [81] and Jungnickel, Menezes and Vanstone [72]. In designing finite field multipliers it is sometime useful to consider weakly self... |

14 |
Primitive normal bases for finite fields
- Schoof
- 1987
(Show Context)
Citation Context ...e group of Fqn. This result was proved by CarlitzsCHAPTER 1. INTRODUCTION 10 [32] in 1952 for qn sufficiently large, by Davenport [38] in 1968 for the case that q is a prime and by Lenstra and Schoof =-=[87]-=- in 1987 for the general case. For the construction of primitive normal bases and primitive elements, see Cohen [34], Hachenberger [60], Shoup [129], Stepanov and Shparlinskiy [131, 132, 133]. An impo... |

14 |
L’invariant de Witt de la forme Tr(x 2
- Serre
- 1984
(Show Context)
Citation Context ... basis coincides with its dual basis then it is said to be self-dual, i.e., a basis {α0,α1,... ,αn−1} is called self-dual if Tr(αiαj) =δi,j. Existence results of self-dual bases can be found in Serre =-=[125]-=- and Kahn [73, 74]. As to the existence of self-dual normal bases, we have the following two theorems. Theorem 1.4.3 Let E be a Galois extension of F of degree n. Ifnis odd then E has a self-dual norm... |

10 |
Tables of Finite Fields
- Alanen, Knuth
- 1964
(Show Context)
Citation Context ...itive integer k such that tqk ≡ t mod e. If t is relatively prime to e, then d is equal to n. Thus if gcd(t, e) = 1 then ft(x) =Ct(x). Several methods of computing Ct(x) are given by Alanen and Knuth =-=[6]-=-, Daykin [39], Rifà and Borrell [113] and Thiong Ly [136]. We will not discuss these methods here. Instead we will show that the coefficients of ft(x) are a unique solution of a system of linear equat... |

10 |
Cours d'algèbre supérieure
- Serret
- 1849
(Show Context)
Citation Context ...ain irreducible over Fq. We assume that p is a prime such that 2a |(p + 1), 2a+1 ∤ (p + 1) with a ≥ 2. Then 2a+1 is the highest power in p2 − 1. We first quote the following result due to J.A. Serret =-=[126]-=-, see also [89, Theorem 3.75].sFACTORING x e − 1 44 Lemma 3.3.4 Let a ∈ F ∗ q with multiplicative order e. Then the binomial xt − a is irreducible in Fq[x] if and only if the integer t ≥ 2 satisfies t... |

9 |
Galois Theory, University of Notre Dame
- Artin
- 1942
(Show Context)
Citation Context ...he probability, κ, that α is normal over Fq satisfies κ ≥ 1/34 if n ≤ q 4 , and κ>(16 log q n) −1 if n ≥ q 4 . A better probabilistic algorithm is based on the following theorem. Theorem 3.1.1 (Artin =-=[9]-=-) Let f (x) be an irreducible polynomial of degree n over Fq and α a root of f (x). Let g(x) = f(x) (x−α)f ′ (α) . Then there are at least q − n(n − 1) elements u in Fq such that g(u) is a normal elem... |

9 |
Bases for finite fields
- Davenport
- 1968
(Show Context)
Citation Context ...basis {α, αq ,... ,αqn−1} such that α also generates the multiplicative group of Fqn. This result was proved by CarlitzsCHAPTER 1. INTRODUCTION 10 [32] in 1952 for qn sufficiently large, by Davenport =-=[38]-=- in 1968 for the case that q is a prime and by Lenstra and Schoof [87] in 1987 for the general case. For the construction of primitive normal bases and primitive elements, see Cohen [34], Hachenberger... |

9 | Dickson Polynomials and Irreducible Polynomials over Finite Fields - Gao, Mullen - 1994 |

9 |
Improving the time complexity of the computation of irreducible and primitive polynomials in finite fields
- Rifà, Borrell
- 1991
(Show Context)
Citation Context ...od e. If t is relatively prime to e, then d is equal to n. Thus if gcd(t, e) = 1 then ft(x) =Ct(x). Several methods of computing Ct(x) are given by Alanen and Knuth [6], Daykin [39], Rifà and Borrell =-=[113]-=- and Thiong Ly [136]. We will not discuss these methods here. Instead we will show that the coefficients of ft(x) are a unique solution of a system of linear equations whose coefficients are from the ... |

9 | Systolic multipliers for finite fields GF(2 m - Yeh, Reed, et al. - 1984 |

8 | Factorization of symmetric matrices and trace-orthogonal bases in finite fields - Seroussi, Lempel - 1980 |