algorithms

The set {1, 2, . . . , N − 1}
b
a · b
Pass
Fail
The one-to-one function b ↦→ a · b shows that at least as many elements fail the test as pass it.
Hey, that was group theory!
For any integer N, the set of all numbers mod N that are relatively prime to N constitute
what mathematicians call a group:
• There is a multiplication operation defined on this set.
• The set contains a neutral element (namely 1: any number multiplied by this remains
unchanged).
• All elements have a well-defined inverse.
This particular group is called the multiplicative group of N, usually denoted Z ∗ N .
Group theory is a very well developed branch of mathematics. One of its key concepts
is that a group can contain a subgroup—a subset that is a group in and of itself. And an
important fact about a subgroup is that its size must divide the size of the whole group.
Consider now the set B = {b : b N−1 ≡ 1 mod N}. It is not hard to see that it is a subgroup
of Z ∗ N
(just check that B is closed under multiplication and inverses). Thus the size of B
must divide that of Z ∗ N . Which means that if B doesn’t contain all of Z∗ N
, the next largest
size it can have is |Z ∗ N |/2.
We are ignoring Carmichael numbers, so we can now assert
If N is prime, then a N−1 ≡ 1 mod N for all a < N.
If N is not prime, then a N−1 ≡ 1 mod N for at most half the values of a < N.
The algorithm of Figure 1.7 therefore has the following probabilistic behavior.
Pr(Algorithm 1.7 returns yes when N is prime) = 1
Pr(Algorithm 1.7 returns yes when N is not prime) ≤ 1 2
We can reduce this one-sided error by repeating the procedure many times, by randomly picking
several values of a and testing them all (Figure 1.8).
Pr(Algorithm 1.8 returns yes when N is not prime) ≤ 1 2 k
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