Primality Testing

1
Primality Testing

• Lecture 35 Primality Testing
• Fermats Theorem
• Primality Testing
• Miller-Rabin

• Lecture 34
• Randomized Algorithms
• Introduction
• Numerical Algorithms
• Monte Carlo Algorithms
• Las Vegas Algorithms
• Average vs. Expected Runtime
• Random vs. Pseudorandom

2
Fermats Little Theorem

• If n is prime, then a n-1 mod n 1
• for any a such that 1 lt a lt n-1

n 3, a 2 then 22 mod 3 1
n 7, a 4 then 46 mod 7 4096 mod 7 1
3
First Prime Number Test
function Fermat (n) a random integer between
1 and n-1 if (expomod (a,n-1,n) 1 then return
true
else return false
4
Recall Expomod
function expomod (a,n,z) computes an mod z i
n r 1 x a mod z while i gt 0 if i is
odd then r rx mod z x x2 mod z i i
?2 return r
5
Converse of Fermats Little Thm.

• Converse of a then b might not hold
• converse if b then a
• a is sufficient, but not necessary, for b.
• What is the converse of Fermats little theorem?

6
Converse of Fermats Little Theorem

• If a n-1 mod n 1 for any a such that 1 lt a
  lt n-1,
• then n is prime.

This is not a true statement.
7
Converse of Fermats Little Thm.

• What does that mean for the algorithm?

8
First Prime Number Test
function Fermat (n) a random integer between
1 and n-1 if (expomod (a,n-1,n) 1 then return
true // n might be prime
else return false // n is not
prime

• Know if n is not prime, but what dont you know?
• Why do you get might be prime?

9
False Witnesses
If Fermat(n) returns true then n might or might
not be prime. Consider 15 414mod151, but 15
clearly isnt prime We call such a number a
in an-1modn1 a false witness that n
is prime How common are false witnesses?
Less than 3.3 of the possible witnesses for n lt
1000 are false. But consider
651,693,055,693,681. Have 99.9965 chance of
picking a false witness (!)
10
Fermat Test

• For any probability d gt 0, there are infinitely
  many numbers for which Fermat discovers
  compositeness with probability less than d.
  i.e. (1-d) chance of false witness.
• Or, Fermat is not p-correct for any p. An
  algorithm is p-correct if it yields a correct
  answer with probability at least p for any
  instance
• Implication repeating Fermat doesnt improve our
  confidence in the result.

11
A Better Test

• n is an odd integer greater than 4.
• s,t so that n-1 2st where t is odd.
• Define B(n) so that a in B(n) if
• at mod n 1, or
• for some i lt s, a(2i) t mod n n - 1
• Then a in B(n) for all a between 2 and n-2 when n
  is prime.

12
A Better Test

• function Btest(a,n)
• s? 0 t? n-1
• repeat
• s? s1 t? floor(t/2)
• until tmod21
• x? expomod(a,t,n)
• if x1 or x n-1 then return true
• for i 1 to s-1 do
• x? x2modn
• if x n-1 then return true
• return false

13
A Better Test

• function Btest(a,n)
• s? 0 t? n-1
• repeat
• s? s1 t? floor(t/2)
• until tmod21
• x? expomod(a,t,n)
• if x1 or x n-1 then return true
• for i 1 to s-1 do
• x? x2modn
• if x n-1 then return true
• return false

Check if 158 belongs to B(289)
288/2 144 s1 144/2 72 s2 72/2 36 s3 36/2
18 s4 18/2 9 s5
x atmodn 1589mod289131
true
14
A Better Test
158 is a strong false witness for 289 172

• function Btest(a,n)
• s? 0 t? n-1
• repeat
• s? s1 t? floor(t/2)
• until tmod21
• x? expomod(a,t,n)
• if x1 or x n-1 then return true
• for i 1 to s-1 do
• x? x2modn
• if x n-1 then return true
• return false

Check if 158 belongs to B(289)
288/2 144 s1 144/2 72 s2 72/2 36 s3 36/2
18 s4 18/2 9 s5
x atmodn 1589mod289131
true
15
A Better Test
158 is a strong false witness for 289 172

• function Btest(a,n)
• s? 0 t? n-1
• repeat
• s? s1 t? floor(t/2)
• until tmod21
• x? expomod(a,t,n)
• if x1 or x n-1 then return true
• for i 1 to s-1 do
• x? x2modn
• if x n-1 then return true
• return false

Check if 158 belongs to B(289)
288/2 144 s1 144/2 72 s2 72/2 36 s3 36/2
18 s4 18/2 9 s5
x atmodn 1589mod289131
true
There are considerably fewer strong false
witnesses than false witnesses
16
A Better Test

• Theorem Consider arbitrary odd n gt 4
• If n is prime, then B(n) a2ltaltn-2
• If n is composite, then B(n) lt (n-9)/4

• If you choose a number between 2 and n-2 with
  equal probability, then there are only (n-9)/4
  opportunities for a false positive.
• What percentage is this? Is there a p for
  which this algorithm is p-correct?

17
A Better Test

• This is called the Miller Rabin test.
• to find out if n is prime, pick a number between
  2 and n-2,
• see if that number is in B(n).
• It is 3/4-correct in the worst case.
• Implication repeating Miller Rabin does increase
  our confidence in the result (!)
• Two witness give .9375-correct answer.