Since the Legendre symbol evaluates to −1, 31 is not a quadratic residue modulo 47. Page 25 Quadratic Reciprocity, VIII Example: Determine whether 357 is a quadratic residue mod 661. Page 26 Quadratic Reciprocity, VIII Example: Determine whether 357 is a quadratic residue mod 661. We want to find (357 661 ) .

Solution: No. We will use quadratic reciprocity. Note that 67 ≡ 31 ≡ 3 mod 4, and 31 and 67 are primes: (31 67 ) = − (67 31 ) = − ( 5 31 ) = − (31 5 ) = − (1 5 ) = −1.

The group of quadratic residues QRN over a Blum integer N = PQ (where P ≡ Q ≡ 3 mod 4) has proven to be a useful group for cryptographic purposes. For example, Rabin [30] proved that computing square roots in this group is equivalent to factoring the modulus N.

We only need to solve, when a number (b) has a square root modulo p, to solve quadratic equations modulo p. Given a number a, s.t., gcd(a, p) = 1; a is called a quadratic residue if x2 = a mod p has a solution otherwise it is called a quadratic non-residue.

For an odd prime p, there are (p+1)/2 quadratic residues (counting zero) and (p-1)/2 non-residues. (The residues come from the numbers 02, 12, 22, , {(p-1)/2}2, these are all different modulo p and clearly list all possible squares modulo p.)…quadratic residue.

Table of quadratic residues

Modulo a prime p, a quadratic residue a has 1 + (a|p) roots (i.e. zero if a N p, one if a ≡ 0 (mod p), or two if a R p and gcd(a,p) = 1.) x2 ≡ 6 (mod 3) has one solution, 0; x2 ≡ 6 (mod 5) has two, 1 and 4. and there are two solutions modulo 15, namely 6 and 9. Solve x2 ≡ 4 (mod 15).

In effect, a quadratic residue modulo p is a number that has a square root in modular arithmetic when the modulus is p . The law of quadratic reciprocity says something about quadratic residues and primes. Quadratic residues are used in the Legendre symbol.

Law of quadratic reciprocity

For example when p = 13 we may take g = 2, so g2 = 4 with successive powers 1,4,3,12,9,10 (mod 13). These are the quadratic residues; to get the quadratic nonresidues multiply them by g = 2 to get the odd powers 2,8,6,11,5,7 (mod 13).

Thus, we conclude that 3 is a quadratic residue modulo p precisely when p = 2, or when p ≡ 1 or 11 (mod 12).

Quadratic residue p-1 modulo p for p primes [duplicate] It seems that for every prime p = (4k + 1), p-1 is a quadratic residue modulo p.

Thus, the number of quadratic residues modulo n cannot exceed n /2 + 1 ( n even) or ( n + 1)/2 ( n odd). The product of two residues is always a residue. Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are ( p + 1)/2 residues (including 0) and ( p − 1)/2 nonresidues, by Euler’s criterion.

Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd). The product of two residues is always a residue.

Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers

If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue. If p ≡ 3 (mod 4) the negative of a residue modulo p is a nonresidue and the negative of a nonresidue is a residue. Prime power modulus. All odd squares are ≡ 1 (mod 8) and thus also ≡ 1 (mod 4).