The group Zp*

Cryptosystems using arithmetic in Zp* include the Diffie-Hellman Key Agreement Protocol and the Digital Signature Algorithm (DSA).

The multiplicative group Zp* uses only the integers between 1 and p - 1 (p is a prime number), and its basic operation is multiplication. Multiplication ends by taking the remainder on division by p; this ensures closure. The multiplicative group Z11* uses the integers from 1 to 10. Multiplication in Z11* finishes by taking the remainder when the result is divided by 11. Here are some examples of multiplication in Z11*:

4 * 7 mod 11 = 28 mod 11 = 6

9 * 5 mod 11 = 45 mod 11 = 1.

Thus in Z11*, 4 * 7 = 6 and 9 * 5 = 1. Notice that both the calculations shown have answers between 1 and 10.
Multiplicative Inverses
Each number x in a multiplicative group has a multiplicative inverse element in the group; that is an integer x^{-1} such that x x^{-1}= 1 in the group. In Z11*, 9-1 = 5 since 9 * 5 mod 11 = 1.

In a multiplicative group, each element must have a multiplicative inverse. Consider the integers modulo the (composite) number 15. It is possible to define multiplication on the numbers from 1 to 14, always finishing with reduction modulo 15. With this system, the number 6 has no inverse, since there is no number y such that 6 * y mod 15 = 1: