# Lecture: Quadratic Reciprocity and the Solovay-Strassen Primality Test

### Background Material

### Reading

### Topics

- Review quadratic residues and Euler's test.
- Legendre symbol and quadratic reciprocity
- Jacobi symbol and quadratic reciprocity
- An algorithm for computing the Jacobi symbol
- Solovay-Strassen primality test

### Maple Worksheet

- jacobi.mw - Maple worksheet illustrating
the legendre and jacobi symbols, quadratic reciprocity and an algorithm
for computing the jacobi symbol, and the Solovay-Strassen primality test.

### Practice Assignment

- Empirically verify that half of the elements of Z_p are quadratic residues and
half are quadratic non-residues for several primes p.
- Verify that for several prime numbers p that all non-zero elements of Z_p satisfy
the Solovay-Strassen test.
- Verify for several composite numbers n that there are elements of Z_n that are
relatively prime to n yet fail the Solovay-Strassen test. Count the number of
elements that are relatively prime to n that fail and satisfy the Solovay-Strassen
test.
- Verify the formulas for the legendre and jacobi symbols for several different
inputs.
- Trace the jacobi algorithm for several inputs.
- Exc. 7.2, 7.3, 7.4, 7.16, 7.17, and 7.20 (see worksheet) from the text.

Created: May 10, 2008 (revised May 7, 2009) by jjohnson AT cs DOT drexel DOT edu