A note on secure multiparty computation via higher residue symbols

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.

The Eleventh Power Residue Symbol

This 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.

On the 8-rank of narrow class groups of ℚ(−4pq), ℚ(−8pq), and ℚ(8pq)

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.

Computational aspects of rational residuosity

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.