The triple of primes are linked modulo 2 ( the R�dei symbol is " 1 ) but are pairwise unlinked modulo 2 ( the Legendre symbols are all 1 ).
22.
This is now known as the Legendre symbol, and an equivalent definition is used today : for all integers " a " and all odd primes " p"
23.
If the required quadratic nonresidue z is to be found by checking if a randomly taken number y is a quadratic nonresidue, it requires ( on average ) 2 computations of the Legendre symbol.
24.
The triple of primes are " linked " modulo 2 ( the R�dei symbol is " 1 ) but are " pairwise unlinked " modulo 2 ( the Legendre symbols are all 1 ).
25.
Just as the quadratic reciprocity law for the Legendre symbol is also true for the Jacobi symbol, the requirement that the numbers be prime is not needed; it suffices that they be odd relatively prime nonunits.
26.
Determining whether " a " is a quadratic residue or nonresidue modulo " n " ( denoted or ) can be done efficiently for prime " n " by computing the Legendre symbol.
27.
This is because given g ^ a and g ^ b, one can efficiently compute the Legendre symbol of g ^ { ab }, giving a successful method to distinguish g ^ { ab } from a random group element.
28.
The case originally considered by C . F . Gauss was the quadratic Gauss sum, for " R " the field of residues modulo a prime number " p ", and & chi; the Legendre symbol.
29.
But since half the numbers between 1 and " n " are nonresidues, picking numbers " x " at random and calculating the Legendre symbol ( " x " | " n " ) until a nonresidue is found will quickly produce one.
30.
The constant implicit in the notation is genus of the curve in question, and so ( Legendre symbol or hyperelliptic case ) can be taken as the degree of " F " . ( More general results, for other values of " N ", can be obtained starting from there .)
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