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Abstract: 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... (Update)
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...is still only known to be possible in deterministic polynomial time if the prime factorization of 2 k Gamma 1 is known. See Shoup [Sho90] It is natural to ask how easy it is to amplify the success probability of the RP[ Phi P] machine M beyond 1=2, without using too many...
.... element of a field of small characteristic or over a field of prime or prime squared order assuming the extended Riemann hypothesis [Sho92] in general it takes time polynomial in the order of a field to find a primitive element in that field or a small extension. At the...
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BibTeX entry: (Update)
V. Shoup. Searching for primitive roots in finite fields. Math. Comp., 58:369--380, 1992. http://citeseer.ist.psu.edu/shoup92searching.html More
@inproceedings{ shoup90searching,
author = "Victor Shoup",
title = "Searching for Primitive Roots in Finite Fields",
booktitle = "{ACM} Symposium on Theory of Computing",
pages = "546-554",
year = "1990",
url = "citeseer.ist.psu.edu/shoup92searching.html" }
Citations (may not include all citations):439 An Introduction to the Theory of Numbers (context) - Hardy, Wright - 1984
86 Finite Fields (context) - Lidl, Niederreiter - 1983
47 Algebraic Number Theory (context) - Cassels, Frohlich - 1967
42 Number Theory (context) - Borevich, Shafarevich - 1966
37 Basic Number Theory (context) - Weil - 1974
35 New algorithms for finding irreducible polynomials over fini.. - Shoup - 1990
23 Finding irreducible polynomials over finite fields (context) - Adleman, Jr - 1986
20 the deterministic complexity of factoring polynomials over f.. - Shoup - 1990
19 The least quadratic nonresidue (context) - Ankeny - 1952
18 Topics in Multiplicative Number Theory (context) - Montgomery - 1971
14 Finding isomorphisms between finite fields (context) - Lenstra - 1991
10 Factoring polynomials and primitive elements for special pri.. (context) - Gathen - 1987
9 A bound for the least prime ideal in the Chebotarev density .. (context) - Lagarias, Montgomery et al. - 1979
8 Riemann hypothesis and finding roots over finite fields (context) - Huang - 1985
7 the least primitive root of a prime (context) - Wang - 1961
6 Factoring a solvable polynomial over a finite field and Gene.. (context) - Evdokimov - 1989
6 Deterministic factorization of polynomials over special fini.. (context) - Bach, Gathen - 1988
5 Smoothness and factoring polynomials over finite fields - Shoup, on et al. - 1989
4 Galois groups and factoring polynomials over finite fields (context) - R'onyai - 1989
4 On primitive roots in finite fields (context) - Davenport - 1937
3 Demonstratio Mathematica (context) - Iwaniec, problem - 1978
3 Factoring polynomials over finite fields (context) - R'onyai - 1988
3 Distribution of primitive roots in a finite field (context) - Carlitz - 1953
2 Factoring polynomials modulo special primes (context) - R'onyai - 1989
2 Character sums and primitive roots in finite fields (context) - Davenport, Lewis - 1963
2 the error term in the linear sieve (context) - Iwaniec - 1971
1 A note on primitive roots in finite fields (context) - Friedlander - 1972
1 Character sums and primitive roots in finite fields (context) - Burgess - 1967
1 Character sums and primitive roots in finite fields (context) - Karacuba - 1968
1 Character sums and primitive roots in algebraic number field.. (context) - Hinz - 1983
1 Polynomial time construction of a finite field (context) - Chistov - 1984
1 On primitive elements in finite fields and on elliptic curve.. (context) - Shparlinsky - 1990
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