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The '''Fermat primality test''' is a [[randomized algorithm|probabilistic]] test to determine if a number is [[probable prime]].
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==Concept==
[[Fermat's little theorem]] states that if ''p'' is prime and <math>1 \le a < p</math>, then
 
:<math>a^{p-1} \equiv 1 \pmod{p}.</math>
 
If we want to test if ''p'' is prime, then we can pick random ''a'''s in the interval and see if the equality holds.  If the equality does not hold for a value of ''a'', then ''p'' is composite.  If the equality does hold for many values of ''a'', then we can say that ''p'' is [[probable prime]].
 
It might be in our tests that we do not pick any value for ''a'' such that the equality fails. Any ''a'' such that
:<math>a^{n-1} \equiv 1 \pmod{n}</math>
when ''n'' is composite is known as a ''Fermat liar''. Vice versa, in this case ''n'' is called [[Fermat pseudoprime]] to base ''a''.
 
If we do pick an ''a'' such that
:<math>a^{n-1} \not\equiv 1 \pmod{n}</math>
then ''a'' is known as a ''Fermat witness'' for the compositeness of ''n''.
 
== Example ==
Suppose we wish to determine if ''n''&nbsp;=&nbsp;221 is prime. Randomly pick 1 ≤ ''a'' < 221, say ''a''&nbsp;=&nbsp;38. We check the above equality and find that it holds:
:<math>a^{n-1} = 38^{220} \equiv 1 \pmod{221}.</math>
Either 221 is prime, or 38 is a Fermat liar, so we take another ''a'', say 26:
:<math>a^{n-1} = 26^{220} \equiv 169 \not\equiv 1 \pmod{221}.</math>
So 221 is composite and 38 was indeed a Fermat liar.
 
==Algorithm and running time==
The algorithm can be written as follows:
:'''Inputs''': ''n'': a value to test for primality; ''k'': a parameter that determines the number of times to test for primality
:'''Output''': ''composite'' if ''n'' is composite, otherwise ''probably prime''
:Repeat ''k'' times:
::Pick ''a'' randomly in the range [1, ''n'' &minus; 1]
::If <math>a^{n-1}\not\equiv1 \pmod n</math>, then return ''composite''
:If composite is never returned: return ''probably prime''
 
Using fast algorithms for [[modular exponentiation]], the running time of this algorithm is [[Big O notation|O]](''k'' × log<sup>2</sup>''n'' × log log ''n'' × log log log ''n''), where ''k'' is the number of times we test a random ''a'', and ''n'' is the value we want to test for primality.
 
==Flaw==
There are infinitely many values of <math>n</math> (known as [[Carmichael number]]s) for which <u>all</u> values of <math>a</math> for which <math>gcd(a, n) = 1</math> are Fermat liars. While Carmichael numbers are substantially rarer than prime numbers,<ref>Erdös' upper bound for the number of Carmichael numbers is lower than the prime number function n/log(n)</ref> there are enough of them that Fermat's primality test is often not used in the above form. Instead, other more powerful extensions of the Fermat test, such as [[Baillie-PSW primality test|Baillie-PSW]], [[Miller-Rabin primality test|Miller-Rabin]], and [[Solovay-Strassen primality test|Solovay-Strassen]] are more commonly used.
 
In general, if <math>n</math> is not a Carmichael number then at least half of all
 
:<math>a\in(\mathbb{Z}/n\mathbb{Z})^*</math>
 
are Fermat witnesses. For proof of this, let <math>a</math> be a Fermat witness and <math>a_1</math>, <math>a_2</math>, ..., <math>a_s</math> be Fermat liars. Then
 
:<math>(a\cdot a_i)^{n-1} \equiv a^{n-1}\cdot a_i^{n-1} \equiv a^{n-1} \not\equiv 1\pmod{n}</math>
 
and so all <math>a \times a_i</math> for <math>i = 1, 2, ..., s</math> are Fermat witnesses.
 
==Applications==
The encryption program [[Pretty Good Privacy|PGP]] uses this primality test in its algorithms. The chance of PGP generating a [[Carmichael number]] is less than 1 in 10<sup>50</sup>, which is more than adequate for practical purposes.
 
==References==
* {{cite book |author=[[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], [[Clifford Stein]] |title=[[Introduction to Algorithms]] |edition=Second Edition |publisher=MIT Press; McGraw-Hill |year=2001 |isbn=0-262-03293-7 |page=889&ndash;890 |chapter=Section 31.8: Primality testing}}
{{reflist}}
 
{{number theoretic algorithms}}
 
[[Category:Primality tests]]
[[Category:Modular arithmetic]]

Latest revision as of 09:26, 22 October 2014

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