The Eleventh Power Residue Symbol

Calculating the Power Residue Symbol and Ibeta

In the recent PhD thesis of Bouw, an algorithm is examined that computes the group structure of the principal units of a p-adic number €eld completion. In the same thesis, this algorithm is used to compute Hilbert norm residue symbols. In the present paper, we will demonstrate two other applications. Œe €rst application is the computation of an important invariant of number €eld completions, ca...

An Efficient Seventh Power Residue Symbol Algorithm

Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In...

A Multi-Value Sequence Generated by Power Residue Symbol and Trace Function over Odd Characteristic Field

The least k-th power non-residue

Let p be a prime number and let k ≥ 2 be an integer such that k divides p − 1. Norton proved that the least k-th power non-residue modp is at most 3.9p log p unless k = 2 and p ≡ 3 (mod 4), in which case the bound is 4.7p log p. With a combinatorial idea and a little help from computing power, we improve the upper bounds to 0.9p log p and 1.1p log p, respectively.

Lam’s power residue addition sets

Article history: Available online xxxx MSC: primary 05B10 secondary 11A15, 11T22, 11T24