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Fast Construction of Irreducible Polynomials over Finite Fields
 J. Symbolic Comput
, 1993
"... The main result of this paper is a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field F q . The algorithm is probabilistic, and is asymptotically faster than previously known algorithms for this problem. It uses an expected number of O~(n 2 + n log q) ..."
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Cited by 48 (6 self)
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The main result of this paper is a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field F q . The algorithm is probabilistic, and is asymptotically faster than previously known algorithms for this problem. It uses an expected number of O~(n 2 + n log q) operations in F q , where the "softO" O~ indicates an implicit factor of (log n) O(1) . In addition, two new polynomial irreducibility tests are described. 1 Introduction 1.1 Statement of main result Let F q be a finite field with q elements, where q is a primepower. A theorem due to Moore (1893) states that for every positive integer n, there exists a field extension F q n , unique up to isomorphism, with q n elements. Such extensions play an important role in coding theory (implementing error correcting codes), cryptography (implementing cryptosystems), and complexity theory (amplifying randomness). In this paper, we consider the algorithmic version of Moore's theorem: how to ...
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 39 (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 ...
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
Removing Randomness From Computational Number Theory
, 1989
"... In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for pri ..."
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Cited by 3 (1 self)
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In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for primality. Even for problems which are known to have deterministic polynomial time algorithms, these algorithms are often not as fast as some probabilistic algorithms for the same problem. Even though probabilistic algorithms are useful in practice, we would like to know, for both theoretical and practical reasons, if randomization is really necessary to obtain the most efficient algorithms for certain problems. That is, we would like to know for which problems there is an inherent gap between the deterministic and probabilistic complexities of these problems. In this research, we consider two problems of a number theoretic nature: factoring polynomials over finite fields and constructing irred...
A Note on Irreducible Polynomials and Identity Testing
"... We show that, given a finite field Fq and an integer d> 0, there is a deterministic algorithm that finds an irreducible polynomial g over Fq in time polynomial in d and log q such that, d d log q < deg(g) < c log p log p where c is a constant. This result follows easily from Adleman and Lenstra’s re ..."
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We show that, given a finite field Fq and an integer d> 0, there is a deterministic algorithm that finds an irreducible polynomial g over Fq in time polynomial in d and log q such that, d d log q < deg(g) < c log p log p where c is a constant. This result follows easily from Adleman and Lenstra’s result [AJ86] on irreducible polynomials over prime fields and Lenstra’s result [Jr.91] on isomorphisms between finite fields. 1 As an application, we show that such construction of irreducible polynomials can be used to build a sample space of coprime polynomials for the AgrawalBiswas [AB03] polynomial identity testing algorithm. 1
Computational Number Theory and Algebra May 16, 2012 Lecture 9
"... In the last class, we mentioned that an irreducible polynomial of degree n over a finite field Fq can be used to generate the extension field Fqn. This gives us a method to construct large finite fields starting from small fields. To give you an example as to where such extension fields are useful, ..."
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In the last class, we mentioned that an irreducible polynomial of degree n over a finite field Fq can be used to generate the extension field Fqn. This gives us a method to construct large finite fields starting from small fields. To give you an example as to where such extension fields are useful, recall that in the ReedSolomon encoding procedure, we need to use a finite field whose size is at least as large as the codeword length. On the other hand, in the list decoding phase we need to factor a bivariate polynomial. Given that bivariate factoring reduces to univariate factoring and that we only know of a deterministic polytime factoring algorithm for lowcharacteristic finite fields, it makes sense to start with a small prime field and extend it suitably to a sufficiently large finite field. In today’s class, we will see how to generate an irreducible polynomial over a finite field in random polynomial time. The topics of discussion for today’s class are: • Generating irreducible polynomials over finite fields, • MillerRabin primality test. 1 Generating irreducible polynomials over finite fields We want to generate an irreducible polynomial of degree n over a finite field Fq. Recall from the last class that irreducibility of a given polynomial can be checked in deterministic polynomial time. Now, if we can show that the density of irreducible polynomials is sufficiently large then we can just pick a random polynomial of degree n and test if it is irreducible. This should yield an irreducible polynomial with high probability (provided the density is large). To make this idea formal, we need to estimate the density of irreducible polynomials of degree n over a finite field Fq.