Quadratic residue information

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer...

is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue)...

number of quadratic residues and non-residues; and The product of two quadratic residues is a residue, the product of a residue and a non-residue is a non-residue...

A quadratic residue code is a type of cyclic code. Examples of quadratic residue codes include the ( 7 , 4 ) {\displaystyle (7,4)} Hamming code over G...

roots, where φ is the Euler's totient function. Quadratic residue: An integer a is a quadratic residue modulo m, if there exists an integer x such that...

{\displaystyle a} and N {\displaystyle N} , whether a {\displaystyle a} is a quadratic residue modulo N {\displaystyle N} or not. Here N = p 1 p 2 {\displaystyle...

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle...

and martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern...

In theoretical computer science and cryptography, a trapdoor function is a function that is easy to compute in one direction, yet difficult to compute...

elements is a quadratic residue if it has a square root in Fq. Otherwise, it is a quadratic non-residue. There are (q − 1)/2 quadratic residues and (q − 1)/2...

or may not be a quadratic residue modulo n. This is because for a to be a quadratic residue modulo n, it has to be a quadratic residue modulo every prime...

then p{\displaystyle p} is a quadratic residue mod q{\displaystyle q} if and only if q{\displaystyle q} is a quadratic residue mod p{\displaystyle p}; by...

The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field...

q) is a quadratic residue, then if a ≡ b2 is a residue, r ≡ a2 ≡ b4 (mod q) is a biquadratic residue, and if a is a nonresidue, −a is a residue, −a ≡ b2...

the prime l {\displaystyle l} is a quadratic residue modulo the prime p {\displaystyle p} there is a quadratic residue code which is a cyclic code of length...

\left({\frac {a}{p}}\right)} is 1 if a is a quadratic residue modulo p; it is –1 if a is a quadratic non-residue modulo p; it is 0 if p divides a. The same...

Since for these primes p, 2p + 1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, and the multiplicative order of 2 mod 2p + 1 must divide...

{an^{2}+bn}{c}}}}. The classical quadratic Gauss sum is the sum g(a, p) = G(a, 0, p). Properties The Gauss sum G(a,b,c) depends only on the residue class of a and b modulo...

≡ m 2 mod p {\displaystyle c\equiv m^{2}{\bmod {p}}} , so c is a quadratic residue modulo p {\displaystyle p} . Then m p 2 ≡ c 1 2 ( p + 1 ) ≡ c ⋅ c...

to 3 (mod 4) (this guarantees that each quadratic residue has one square root which is also a quadratic residue), and should be safe primes with a small...

1 modulo 8. Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue modulo p, that is, there is integer a such that p | a 2 − 2. {\displaystyle...