residue symbol
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2021 ◽  
Vol 15 (1) ◽  
pp. 284-297
Author(s):  
Ignacio Cascudo ◽  
Reto Schnyder

Abstract We generalize a protocol by Yu for comparing two integers with relatively small difference in a secure multiparty computation setting. Yu's protocol is based on the Legendre symbol. A prime number p is found for which the Legendre symbol (· | p) agrees with the sign function for integers in a certain range {−N, . . . , N} ⊂ ℤ. This can then be computed efficiently. We generalize this idea to higher residue symbols in cyclotomic rings ℤ[ζr ] for r a small odd prime. We present a way to determine a prime number p such that the r-th residue symbol (· | p) r agrees with a desired function f : A → { ζ r 0 , … , ζ r r − 1 } f:A \to \left\{ {\zeta _r^0, \ldots ,\zeta _r^{r - 1}} \right\} on a given small subset A ⊂ ℤ[ζr ], when this is possible. We also explain how to efficiently compute the r-th residue symbol in a secret shared setting.


2020 ◽  
Vol 15 (1) ◽  
pp. 111-122
Author(s):  
Marc Joye ◽  
Oleksandra Lapiha ◽  
Ky Nguyen ◽  
David Naccache

AbstractThis paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclo-tomic field $ \mathbb{Q}\left( {{\zeta }_{11}} \right), $where 11 is a primitive 11th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.


Author(s):  
Xiaopeng Zhao ◽  
Zhenfu Cao ◽  
Xiaolei Dong ◽  
Jun Shao ◽  
Licheng Wang ◽  
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2018 ◽  
Vol 14 (08) ◽  
pp. 2165-2193 ◽  
Author(s):  
Djordjo Z. Milovic

Let [Formula: see text]. We study the [Formula: see text]-part of the narrow class group in thin families of quadratic number fields of the form [Formula: see text], where [Formula: see text] are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order [Formula: see text]. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.


2018 ◽  
Vol 14 (03) ◽  
pp. 631-645
Author(s):  
Markus Hittmeir

In this paper, we consider an extension of Jacobi’s symbol, the so-called rational [Formula: see text]th power residue symbol. In Sec. 3, we prove a novel generalization of Zolotarev’s lemma. In Secs. 4–6, we show that several hard computational problems are polynomial-time reducible to computing these residue symbols, such as getting nontrivial information about factors of semiprime numbers. We also derive criteria concerning the Quadratic Residuosity Problem.


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