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15
Searching for Primitive Roots in Finite Fields
, 1992
"... Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). ..."
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Cited by 40 (3 self)
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Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n O(1) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p) assuming the ERH. Appeared in Mathematics of Computation 58, pp. 369380, 1992. An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 546554. 1980 Mathematics Subject Classification (1985 revision): 11T06. 1. Introduction Consider the problem of finding a primitive ...
Normal Bases over Finite Fields
, 1993
"... 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 repr ..."
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Cited by 9 (0 self)
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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...
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 8 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
On the function field sieve and the impact of higher splitting probabilities: Application to discrete logarithms in f 2
, 1971
"... Abstract. In this paper we propose a binary field variant of the JouxLercier mediumsized Function Field Sieve, which results not only in complexities as low as Lqn(1/3, 2/3) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms o ..."
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Cited by 7 (1 self)
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Abstract. In this paper we propose a binary field variant of the JouxLercier mediumsized Function Field Sieve, which results not only in complexities as low as Lqn(1/3, 2/3) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one elements. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite field with 2 1971 elements. 1
BlackBox Extension Fields and the Inexistence of FieldHomomorphic OneWay Permutations
"... The blackbox field (BBF) extraction problem is, for a given field�, to determine a secret field element hidden in a blackbox which allows to add and multiply values in�in the box and which reports only equalities of elements in the box. This problem is of cryptographic interest for two reasons. Fi ..."
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Cited by 6 (0 self)
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The blackbox field (BBF) extraction problem is, for a given field�, to determine a secret field element hidden in a blackbox which allows to add and multiply values in�in the box and which reports only equalities of elements in the box. This problem is of cryptographic interest for two reasons. First, for ���Ôit corresponds to the generic reduction of the discrete logarithm problem to the computational DiffieHellman problem in a group of prime orderÔ. Second, an efficient solution to the BBF problem proves the inexistence of certain fieldhomomorphic encryption schemes whose realization is an interesting open problems in algebrabased cryptography. BBFs are also of independent interest in computational algebra. In the previous literature, BBFs had only been considered for the prime field case. In this paper we consider a generalization of the extraction problem to BBFs that are extension fields. More precisely we discuss the representation problem defined as follows: For given generators��������algebraically generating a BBF and an additional elementÜ, all hidden in a blackbox, expressÜalgebraically in terms of ��������. We give an efficient algorithm for this representation problem and related problems for fields with small characteristic (e.g.���Òfor someÒ). We also consider extension fields of large characteristic and show how to reduce the representation problem to the extraction problem for the underlying prime field. These results imply the inexistence of fieldhomomorphic (as opposed to only grouphomomorphic, like RSA) oneway permutations for fields of small characteristic.
Constructing Normal Bases in Finite Fields
 J. Symbolic Comput
, 1990
"... This paper addresses the question: how can we find a normal element efficiently? More generally, we consider how to find an element of any given additive order. Hensel (1888) pioneered the study of normal bases for finite fields and proved that they always exist. We use his algorithm in Section 2. E ..."
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Cited by 5 (0 self)
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This paper addresses the question: how can we find a normal element efficiently? More generally, we consider how to find an element of any given additive order. Hensel (1888) pioneered the study of normal bases for finite fields and proved that they always exist. We use his algorithm in Section 2. Eisenstein (1850) had already noted that normal bases always exist. Hensel, and also Ore (1934), determine exactly the number of these bases, and Ore develops the more general concept of additive order. Ore's approach is developed into more constructive proofs of the normal basis theorem in several textbooks (for example, van der Waerden 1966, Section 67, and Albert 1956, Section 4.15); these all use some linear algebra calculations. Schwarz (1988) has given a new proof along these lines, and several recent papers have translated this approach into algorithms. Sidel'nikov (1988) deals with the case where n divides one of p (the characteristic of F q ), q + 1, or
Communication Complexity of Key Agreement on Limited Ranges (Extended Abstract)
, 1994
"... This paper studies a variation on classical keyagreement and consensus problems in which the set S of possible keys is the range of a random variable that can be sampled. We give tight upper and lower bounds of dlog 2 ke bits on the communication complexity of agreement on some key in S, using a fo ..."
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Cited by 3 (1 self)
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This paper studies a variation on classical keyagreement and consensus problems in which the set S of possible keys is the range of a random variable that can be sampled. We give tight upper and lower bounds of dlog 2 ke bits on the communication complexity of agreement on some key in S, using a form of Sperner's Lemma, and give bounds on other problems. In the case where keys are generated by a probabilistic polynomialtime Turing machine, agreement is shown to be possible with zero communication if every fully polynomialtime approximation scheme (fpras) has a certain symmetrybreaking property. 1 Introduction A fundamental problem in key agreement between two parties, commonly called "Alice" and "Bob," is for Alice to communicate some string w to Bob over an expensive, noisy, and/or insecure channel. Most work allows w to be any given string, making no assumptions about its source, or considers w to be uniformly generated among strings of some length n. We study cases in which w...
Probabilistic Construction of Normal Basis.
, 1998
"... Let Fq be the finite field with q elements. A normal basis polynomial f # Fq [x] of degree n is an irreducible polynomial, whose roots form a (normal) basis for the field extension Fq n : Fq . We show that a normal basis polynomial of degree n can be found in expected time O(n 2+# · log(q) + ..."
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Cited by 2 (0 self)
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Let Fq be the finite field with q elements. A normal basis polynomial f # Fq [x] of degree n is an irreducible polynomial, whose roots form a (normal) basis for the field extension Fq n : Fq . We show that a normal basis polynomial of degree n can be found in expected time O(n 2+# · log(q) + n 3+# ), when an arithmetic operation and the generation of a random constant in the field Fq cost unit time. Given some basis B = {#1 , #2 , ..., #n} for the field extension Fq n : Fq together with an algorithm for multiplying two elements in the B representation in time O(n # ), we can find a normal basis for this extension and express it in terms of B in expected time O(n 1+#+# · log(q) + n 3+# ). CR Categories: F.2.1. 1991 Mathematics Subject Classification: Primary 11Y16; Secondary 11T30. Related Work. [BDS90] give a probabilistic construction of a normal basis for F q n : F q for restricted values of q and n. They use that the ground field F q can have at most n(n  1...
Specific Irreducible Polynomials with Linearly Independent Roots over Finite Fields
"... In this paper we give several families of specific irreducible polynomials with the following property: if f(x) is one of the given polynomials of degree n over a finite field F q and # is a root of it, then # # F q n is normal over every intermediate field between F q n and F q . Here by # # F q ..."
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In this paper we give several families of specific irreducible polynomials with the following property: if f(x) is one of the given polynomials of degree n over a finite field F q and # is a root of it, then # # F q n is normal over every intermediate field between F q n and F q . Here by # # F q n being normal over a subfield F q we mean that the algebraic conjugates #, # are linearly independent over F q . The degrees of the given polynomials are of the form 2 i where r 1 , r 2 , ...,r u are distinct odd prime factors of q  1 and k, l 1 ,...,l u are arbitrary positive integers. For example, we prove that, for a prime p # 3 mod 4, if x  bx  1 # F p [x]is irreducible with b #= 2 then the polynomial  x has the described property over F p for every integer k # 0. We will also show how to e#ciently compute the required b # F p .
Galois Field Library: Reference Manual
, 1998
"... Galois Field Library (GFL) is a portable generalpurpose computational library of functions written in C for working over finite fields. The library provides a comprehensive treatment of operations in prime fields and their arbitrary finite extensions. Application programmers should find this librar ..."
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Galois Field Library (GFL) is a portable generalpurpose computational library of functions written in C for working over finite fields. The library provides a comprehensive treatment of operations in prime fields and their arbitrary finite extensions. Application programmers should find this library useful for developing programs in the areas of publickey cryptography, error control coding and combinatorial design. This technical report is a reference manual of GFL. It provides an exhaustive listing of all the features of the library  namely the new data structures and macro definitions introduced in the header files of the library and the prototypes of all GFL library calls. KEY WORDS: Finite fields Data structures Algorithms Library 1 Introduction Galois Field Library (GFL) is a portable generalpurpose computational library of functions written in C for working over finite fields (also called Galois fields). GFL provides routines for field arithmetic and for manipulation of uni...