The least k-th power non-residue

Treviño, Enrique

doi:10.1016/j.jnt.2014.10.019

Cited by 14 publications

(18 citation statements)

References 12 publications

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“…We now aim at bounding each of the sums in (6), which we denote by S 1 , S 2 , S 3 . We have |S 1 | ≤ X −1 by Claim 3.1 in [21]. This could be improved, but will suffice for our purposes.…”

Section: A Collection Of Intervalsmentioning

confidence: 97%

“…When n > 2, the counting problem is much more difficult. Treviño (see Lemma 2.1 of [21]) shows that the number of exceptions is bounded above by the quantity…”

Section: A Character Sum Estimatementioning

confidence: 99%

“…The first part of the following proposition is due to Treviño, following Burgess, Norton, and Booker (see [21]); the second part is a refinement in a special case. Proposition 1.…”

Section: A Character Sum Estimatementioning

confidence: 99%

“…The collection of intervals given in the following proposition is a variation on those that appear in a 1957 paper of Burgess (see [1]) and are standard in the study of explicit bounds for character non-residues (see, for example, [14,15,12,21]). Explicit upper and lower bounds on the number of integer points in these intervals will be essential for our results.…”

Section: A Collection Of Intervalsmentioning

confidence: 99%

“…Treviño [21] gives a bound on this between his equations (15) and (16). We can do a little better with (7) using partial summation; namely |T 3 | ≤ 6π −2 log X + 2.…”

Section: A Collection Of Intervalsmentioning

confidence: 99%

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Explicit upper bounds on the least primitive root

McGown

Trudgian

2019

Proc. Amer. Math. Soc.

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We give a method for producing explicit bounds on g(p), the least primitive root modulo p. Using our method we show that g(p) < 2r 2 rω(p−1) p 1 4 + 1 4r for p > 10 56 where r ≥ 2 is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of p. For example, our method also allows us to show that

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Section: A Collection Of Intervalsmentioning

confidence: 97%

“…When n > 2, the counting problem is much more difficult. Treviño (see Lemma 2.1 of [21]) shows that the number of exceptions is bounded above by the quantity…”

Section: A Character Sum Estimatementioning

confidence: 99%

“…The first part of the following proposition is due to Treviño, following Burgess, Norton, and Booker (see [21]); the second part is a refinement in a special case. Proposition 1.…”

Section: A Character Sum Estimatementioning

confidence: 99%

Section: A Collection Of Intervalsmentioning

confidence: 99%

“…Treviño [21] gives a bound on this between his equations (15) and (16). We can do a little better with (7) using partial summation; namely |T 3 | ≤ 6π −2 log X + 2.…”

Section: A Collection Of Intervalsmentioning

confidence: 99%

See 3 more Smart Citations

Explicit upper bounds on the least primitive root

McGown

Trudgian

2019

Proc. Amer. Math. Soc.

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Fast Secure Comparison for Medium-Sized Integers and Its Application in Binarized Neural Networks

Abspoel

Bouman

Schoenmakers

et al. 2019

Topics in Cryptology – CT-RSA 2019

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In 1994, Feige, Kilian, and Naor proposed a simple protocol for secure 3-way comparison of integers a and b from the range [0, 2]. Their observation is that for p = 7, the Legendre symbol (x | p) coincides with the sign of x for x = a−b ∈ [−2, 2], thus reducing secure comparison to secure evaluation of the Legendre symbol. More recently, in 2011, Yu generalized this idea to handle secure comparisons for integers from substantially larger ranges [0, d], essentially by searching for primes for which the Legendre symbol coincides with the sign function on [−d, d]. In this paper, we present new comparison protocols based on the Legendre symbol that additionally employ some form of error correction. We relax the prime search by requiring that the Legendre symbol encodes the sign function in a noisy fashion only. Practically, we use the majority vote over a window of 2k + 1 adjacent Legendre symbols, for small positive integers k. Our technique significantly increases the comparison range: e.g., for a modulus of 60 bits, d increases by a factor of 2.9 (for k = 1) and 5.4 (for k = 2) respectively. We give a practical method to find primes with suitable noisy encodings. We demonstrate the practical relevance of our comparison protocol by applying it in a secure neural network classifier for the MNIST dataset. Concretely, we discuss a secure multiparty computation based on the binarized multi-layer perceptron of Hubara et al., using our comparison for the second and third layers.

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