﻿ Jacobi-Symbol

# Jacobi-Symbol

The Jacobi symbol, named after Carl Gustav Jacob Jacobi, is a generalization of the Legendre symbol. The Jacobi symbol can be generalized to the Kronecker symbol again. The notation is the same as that of the Legendre symbol:

To distinguish between the Legendre symbol and the Jacobi symbol, it also writes L ( A, P ) and J ( a, n).

It need not be prime, in contrast to the Legendre symbol, but it must be greater than 1 in an odd number act. For the Jacobi symbol as the Legendre symbol all integers are allowed.

## N is a prime number

If a prime number is, the Jacobi symbol behaves exactly as the Legendre symbol:

## N is not a prime number

If the prime factorization of, so we define

Example:

Note: If no prime, are the Jacobi symbol does not indicate whether a quadratic residue modulo (like the Legendre symbol ). A necessary condition that a quadratic residue modulo is, however, that the Jacobi symbol is not equal.

## General definition

Generally, the Jacobi symbol J (a, n ) over a character of the group defined:

It is the following function:

Is any system half modulo, as the value of is independent of the choice of the semiconductor system. denotes the correction factor of and respect:

## Closed representation

The following formula is a closed representation of the value of the Jacobi symbol:

However, to effectively calculate this formula is very useful, since it has, for more quickly many factors.

## Efficient calculation of the Jacobi symbol

In most cases where you need to calculate the Jacobi symbol, so the Solovay -Strassen test, you do not have prime factorization of the number n in J (a, n ) so that the Jacobi symbol is not the Legendre can be traced back symbol. In addition, the above mentioned closed-form representation for bigger is not efficient enough.

However, there are a few rules of calculation with which J ( a, n) can be determined efficiently. These rules are, among others from the quadratic reciprocity law, which has its validity also for the Jacobi symbol.

The most important principle is the following: For all odd integers greater than 1:

This rule is the law of quadratic reciprocity for the Jacobi symbol. With their help, as well as a few other calculation rules can be J (a, b ) for all a, b determine with relatively little effort, which is comparable to the Euclidean algorithm for finding the greatest common divisor. The calculation rules that are additionally needed are the following:

The above rule follows from the definition of the Jacobi symbol of the character. The Jacobi symbol, the counter is only a representative of the group; therefore it does not matter which representatives are selected.

• ( Multiplicativity in the numerator )
• ( Multiplicativity in the denominator )

As an example, J (127, 703) is to be determined:

Since one may choose the representatives in the numerator free, this is equal to

As 2 to 127 is relatively prime, J ( 2, 127) is not 0, and thus ensure J (2, 127 ) 2 = 1 Thus, this factor is eliminated and we obtain:

For the 2 in the numerator there is a closed formula, therefore we obtain finally:

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