Lecture: Quadratic Reciprocity and the Solovay-Strassen Primality Test

Background Material

• Lecture on RSA
• Lecture on Pseudoprimes, quadratic residues, and probabalistic primality testing

Topics

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

Maple Worksheet

1. 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

1. Empirically verify that half of the elements of Z_p are quadratic residues and half are quadratic non-residues for several primes p.
2. Verify that for several prime numbers p that all non-zero elements of Z_p satisfy the Solovay-Strassen test.
3. 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.
4. Verify the formulas for the legendre and jacobi symbols for several different inputs.
5. Trace the jacobi algorithm for several inputs.
6. Exc. 7.2, 7.3, 7.4, 7.16, 7.17, and 7.20 (see worksheet) from the text.