Calculating the Power Residue Symbol and Ibeta

The Jacobi Symbol and a Method of Eisenstein for Calculating It

We present an exposition of the basic properties of the Jacobi symbol, with a method of calculating it due to Eisenstein. Fix a prime p. For an integer a relatively prime to p the Legendre symbol is defined by (a/p) = 1 if a is a quadratic residue (mod p) and (a/p) =−1 if a is a quadratic nonresidue (mod p). We recall Euler’s theorem that (a/p) ≡ a(p−1)/2 (mod p). We have the famous Law of Quad...

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...

Optimal Power Management to Minimize SER in Amplify and-Forward Relay Networks

This paper studies optimal power allocation to minimize symbol error rate (SER) of amplify-and-forward cooperative diversity networks. First, we analytically solve optimal power allocation problem to minimize SER for three different scenarios, namely, multi-branch single-relay, single-branch multi-relay and multi-branch multi-relay cooperative diversity networks, all subject to a given total re...

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

Myosin Ibeta is located at tip link anchors in vestibular hair bundles.

Recent studies have suggested that myosin Ibeta mediates the adaptation of mechanoelectrical transduction in vestibular hair cells. An important prediction of this hypothesis is that myosin Ibeta should be found in the side insertional plaque, an osmiophilic hair bundle structure that anchors tip links and is thought to house the adaptation motor. To determine whether myosin Ibeta was situated ...