Searching for Primitive Roots in Finite Fields

Shoup, Victor

doi:10.2307/2153041

Cited by 51 publications

(62 citation statements)

References 5 publications

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“…Assuming the Grand Riemann Hypothesis (GRH), Wang [18] also proved that g(p) m 6 log 2 p . In 1992, utilizing a combinatorial sieve due to Iwaniec [10] and certain form of character sums, Shoup [15] showed under GRH, g(p) m 4 (log m + 1) 4 log 2 p .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the least primitive root in number fields

Wang

Гонг

2010

Sci. China Math.

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Let K be an algebraic number field and O K its ring of integers. For any prime ideal p, the group (O K /p) * of the reduced residue classes of integers is cyclic. We call any element of a generator of the group (O K /p) * a primitive root modulo p. Stimulated both by Shoup's bound for the rational improvement and Wang and Bauer's generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal p under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the least primitive root in number fields

Wang

Гонг

2010

Sci. China Math.

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“…Recall the following results about finite fields, from [20] and [21]. The following is an explicit construction which approximates any desired edge density p(n) up to an additive term of (n) < Θ(1/n), and achieves D(n) which is optimal up to a constant.…”

Section: If M Is B-smooth Then a Runs In Time Poly(log N B)mentioning

confidence: 99%

Efficiently Constructible Huge Graphs That Preserve First Order Properties of Random Graphs

Naor

Nussboim

Tromer

2005

Theory of Cryptography

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Abstract. We construct efficiently computable sequences of randomlooking graphs that preserve properties of the canonical random graphs G(2 n , p(n)). We focus on first-order graph properties, namely properties that can be expressed by a formula φ in the language where variables stand for vertices and the only relations are equality and adjacency (e.g. having an isolated vertex is a first-order property ∃x∀y(¬edge(x, y))). Random graphs are known to have remarkable structure w.r.t. first order properties, as indicated by the following 0/1 law: for a variety of choices of p(n), any fixed first-order property φ holds for G(2 n , p(n)) with probability tending either to 0 or to 1 as n grows to infinity.We first observe that similar 0/1 laws are satisfied by G(2 n , p(n)) even w.r.t. sequences of formulas {φn} n∈N with bounded quantifier depth, depth(φn) ≤ n lg(1/p(n)) . We also demonstrate that 0/1 laws do not hold for random graphs w.r.t. properties of significantly larger quantifier depth. For most choices of p(n), we present efficient constructions of huge graphs with edge density nearly p(n) that emulate G(2 n , p(n)) by satisfying Θ( n lg(1/p(n)) )-0/1 laws. We show both probabilistic constructions (which also have other properties such as K-wise independence and being computationally indistinguishable from G(N, p(n)) ), and deterministic constructions where for each graph size we provide a specific graph that captures the properties of G(2 n , p(n)) for slightly smaller quantifier depths.

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“…If v = 0, we can slightly change V by using several new decompositions of the form (15). Thus, we can restrict ourselves to the case v = 0.…”

Section: The Principal Ideal Problemmentioning

confidence: 99%

“…Let Λ 0 be the s 1 ×r 1 matrix whose rows are the vectors l 1 (cα−d) ∈ R r1 , where c, d range over all pairs used in (15). Each coordinate of l 1 (cα − d) is determined with an accuracy of ≈ B 1 binary digits.…”

Section: The Principal Ideal Problemmentioning

confidence: 99%

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Special prime numbers and discrete logs in finite prime fields

Semaev¹

2000

Math. Comp.

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) 1/3 = 1.9229 · · · . Using p ∈ A rather than general p does not enhance the performance of Schirokauer's algorithm. The definition of the set A and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some p.Let F p be a finite field of prime order p, and a ∈ F p its primitive element. The discrete log problem in F p is as follows: given a nonzero b ∈ F p , find the residue y(mod p − 1) for y such that a y = b in F p . The security of several cryptographic systems depends on the difficulty of computing discrete logs [1,2]. The best known algorithm for computing discrete logs in as p → ∞, 0 < α < 1, and 0 < β. This method is an adaptation of the popular Number Field Sieve algorithm (NFS), which has been used previously for factorization. It comes from the Gaussian integers method derived in [4] for computing discrete logs in F p . The NFS algorithm is based on the congruencewhere f (x) is an irreducible polynomial in Z[x] and m ∈ Z. The main parameter of the method is k = deg f (x); the other parameters, such as m and the coefficients of f (x), are bounded by p 1/k in absolute value. There exists p for which the coefficients of f (x) are no larger than p o(1/k) in absolute value. For example, let ab t + c ≡ 0(mod p) for |a|, |b|, |c| = O(1) as t → ∞. Then we have (1) with f (x) = ax k + cb t0 , and m = b (t+t0)/k , where t ≡ −t 0 (mod k) and 0 ≤ t 0 < k.

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Searching for Primitive Roots in Finite Fields

Cited by 51 publications

References 5 publications

On the least primitive root in number fields

On the least primitive root in number fields

Efficiently Constructible Huge Graphs That Preserve First Order Properties of Random Graphs

Special prime numbers and discrete logs in finite prime fields

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