Download Lecture 6 (powerpoint): finding a gigantic prime number

Primality Testing
Patrick Lee
12 July 2003
(updated on 13 July 2003)
Finding a Prime Number
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Finding a prime number is critical for publickey cryptosystems, such as RSA and DiffieHellman.
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Naïve approach:
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Randomly pick a number n. Try if n is divided by
2, 3, 5, 7, …., p, where p is the largest prime
number less than or equal to the square root of n.
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Computationally expensive.
You need to pre-obtain all small prime numbers.
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Introduction to Number Theory
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Number theory: modular arithmetic on a finite set
of integers
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Most of the randomized algorithms starts by
choosing a random number from some domain
and then works deterministically from there on. We
hope that with high probability the chosen number
has some desirable properties.
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Goal: Given a number n, the desired complexity is
O(logn), i.e., polynomial in the length of n.
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Computing GCD
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gcd(a, b): greatest common divisor of (a,b)
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Euclid’s algorithm:
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a and b are co-prime iff gcd(a,b) = 1
Finding gcd(a,b)
for a>b, gcd(a,b) = gcd(b, a mod b)
Extended Euclid’s:
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Finding gcd d and numbers x and y such that
d=ax+by
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Groups
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Additive Group:
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Zn = {0, 1, …, n-1} forms a group under addition
modulo n.
Multiplicative Group:
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Zn* = {x | 1 <= x < n and gcd(x,n) = 1} forms a
group under multiplication modulo n.
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For prime p, Zp* includes all elements [1,p-1].
E.g., Z6* = {1, 5}
E.g., Z7* = {1, 2, 3, 4, 5, 6}
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Chinese Remainder Theorem (CRT)
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Given n1, n2,…, nk are pairwise co-prime.
There exists a unique r, r in [0, n = n1n2…nk),
satisfying
r = ri mod ni
for any sequence {r1,..,rk}, where ri in [0, ni).
E.g.,
r = 2 (mod 3)
r = 3 (mod 5)
r = 2 (mod 7)
We have r = 23, unique in [0,105).
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Euler phi Function: phi(.)
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phi(n) = |Zn*|
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Theorem: if n= p1e1p2e2…pkek,
phi(n) = (p1-1)p1e1 - 1...(pk-1)pkek – 1
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e.g., phi(p) = p–1 for prime p
e.g., if n = pq, phi(n) = (p-1)(q-1)
If we know phi(n), we can factorize n.
Euler’s Theorem: for all n and x in Zn*
xphi(n) = 1 (mod n)
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For any prime p, xp-1 = 1 (mod p) for all x in [1, p-1].
(Fermat’s Little Theorem).
If xn-1 <> 1, n is not prime (e.g., 45 mod 6 = 4).
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Order and Generator
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ord(x): smallest t such that xt = 1 mod n
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Generator: an element whose order = group size.
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E.g., in Z11*, ord(3) = 5, ord(2) = 10
E.g., 3 is the generator of Z7*
Subgroup: generated from an element of order t < phi(n)
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{1,3,32=9,33=5,34=4} = {1,3,4,5,9} is a subgroup of Z11*
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A group is cyclic if it has a generator.
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For any prime p, the group Zp* is cyclic, i.e, every Zp* has a
generator, say g.
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Zp* = {1, g, g2, g3, …, gp-2}
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Group Size
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Subgroup size divides group size (for all n)
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Group size = phi(n)
We use an element of order t < phi(n) as the generator
of the subgroup, (say 2 in Z7*).
The subgroup spans t elements.
For x in subgroup, we observe t has to divide phi(n) so
that xtk = xphi(n) = 1, for some integer k. You can prove it
by contradiction by assuming t does not divide phi(n).
E.g., H = {1, 3, 4, 5, 9} is a subgroup of Z11*, |H| dividies
|Z11*|.
This proposition applies to all n (prime / composite).
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Quadratic Residue
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y is a quadratic residue (mod n) if there
exists x in Zn* such that x2 = y (mod n)
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i.e., y has a square root in Zn*
Claim: For any prime p, every quadratic
residue has exactly two square roots x, -x
mod p.
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Proof: if x2 = u2 (mod p), then (x-u)(x+u) = 0
(mod p), so either p divides x-u (i.e., x=u), or p
divides x+u (i.e., x=-u).
It implies if x2 = 1 (mod p), x = 1 or -1.
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Quadratic Residue (cont’d)
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Theorem: For any prime p, and g is
generator, gk is a quadratic residue iff k is
even.
Given Zp* = {1, g, g2, g3, …, gp-2}
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Even powers of g are quadratic residues
Odd powers of g are not quadratic residues
Legendre symbol:
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[a/p] = 1 if a is a quadratic residue mod p, and 1 if a is not a quadratic residue mod p.
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Quadratic Residue (cont’d)
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Theorem: For prime p and a in Zp*, [a/p] = a(p-1)/2
(mod p).
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Zp* is cyclic, a = gk for some k.
If k is even, let k = 2m, a(p-1)/2 = g(p-1)m = 1.
If k is odd, let k = 2m+1, a(p-1)/2 = g(p-1)/2 = -1. Reasons:
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This is a square root of 1.
g(p-1)/2 <> 1 since ord(g) <> (p-1)/2.
But 1 has two square roots. Thus, the only solution is -1.
If n is prime, a(n-1)/2= 1 or -1. If we find a(n-1)/2 is
not 1 and -1, n is composite.
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Ideas of Primality Testing
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Idea 1:
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If xn-1 mod n <> 1, n is definitely composite.
If xn-1 mod n = 1, n is probably prime.
Idea 2:
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If x(n-1)/2 mod n <> {1,-1}, n is definitely
composite.
If x(n-1)/2 mod n = {1,-1}, n is probably prime.
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Simple Primality Testing Alg.
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Repeat k times:
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Pick a in {2,...,n-1} at random.
If gcd(a,n) != 1, then output COMPOSITE.
[this is actually unnecessary but conceptually helps]
If a(n-1)/2 is not congruent to +1 or -1 (mod n), then output
COMPOSITE.
Now, if we ever got a "-1" above output
"PROBABLY PRIME" else output "PROBABLY
COMPOSITE".
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Error of the Simple Alg.
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The alg is BPP with error probability 1/2k.
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If n is prime, half of them makes a(n-1)/2 = 1. Prob. error in
each iteration is ½.
If n is composite, error occurs if n is claimed to be
“PROBABLY PRIME”. We use the key lemma.
Key Lemma: Let n be an odd composite, not a
prime power, and let t=(n-1)/2. If there exists a in
Zn* such that at = -1 (mod n), then at most half of
the x's in Zn* have xt = {-1,+1} (mod n).
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Error of the Simple Alg. (cont’d)
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Let S = {x in Zn* | xt = 1 or -1} (let t = (n-1)/2).
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We’d like to show S is a proper subgroup of Zn*.
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S is a subgroup of Zn* since it's closed under multiplication
(xt)(yt) = (xy)t.
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Find b in Zn* but not in S.
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Let n = qr, where q and r are co-prime.
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Using the CRT notation, let b = (a,1), denoting b=a (mod q),
b=1 (mod r). CRT assures the existence of b.
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Thus, bt = (at, 1t) = (-1, 1), implying b <> 1 and -1, since 1 =
(1, 1) and -1 = (-1,-1).
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S is a proper subgroup. Since the subgroup size divides
the group size, |S| <= ½ |Zn*|.
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Case of Prime-Power Composites
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Key Lemma doesn’t apply if n is a prime-power. However, it
doesn’t matter since it cannot pass the test of step (3), i.e.,
we are sure that a(n-1)/2 <> 1,-1 mod n for all a.
Proof (assume all operations are mod n):
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Write n = pe, where p is prime.
Consider an-1, which is equal to ape-1.
Note that phi(n) = pe-1(p-1) = pe-pe-1, according to the theorem in
slide 7.
e
e-1
e-1
ap -1 = aphi(n)+p -1 = ap -1 (by Euler’s Theorem)
e
Recursively, we get ap -1 = a-1.
Since a<>1, a-1 <> 1. We have an-1 <> 1, and its square root is not 1
and -1.
Thus, if n is prime-power, it does not pass the test case in step (3).
We can safely ignore the case of prime-powers in the Key Lemma.
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Miller-Rabin Algorithm
1)
pick a in {2,...,n-1} at random.
2)
If an-1 != 1 (mod n), then output COMPOSITE
3)
Let n-1 = 2r * B, where B is odd.
4)
Compute aB, a2B, ..., an-1 (mod n).
5)
6)
7)
If we found a non {-1,+1} root of 1 in the above
list, then
output COMPOSITE.
else output POSSIBLY PRIME.
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Error of MR Algorithm
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It is RP.
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For prime n, the algorithm always returns prime.
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For non-Carmichael composite n, the algorithm
returns prime with probability at most ½ in each
iteration (i.e., step 2 detects compositeness with
probability at least ½).
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Carmichael number: a composite n such that for all
a in Zn*, an-1 = 1 mod n. (e.g., 561, 1729)
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Error of MR Algorithm (Proof)
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Let Fn = {x in Zn* | xn-1 = 1 mod n}, the set of
elements that do not violate Fermat’s theorem.
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Lemma: Let n be a composite non-Carmichael
number. Then |Fn| <= ½ |Zn*|.
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Clearly, Fn <> Zn* .
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Fn forms a group.
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There exists a such that an-1 <> 1 mod n.
It is closed under multiplication (trivial proof!)
Fn is a proper subgroup of Zn*. |Fn| divides |Zn*|, and |Fn|
is strictly less than |Zn*|.
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Detecting Carmichael Numbers
aB ,
rB
2
a
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Computing
...,
(mod n), where B =(n-1)/2r,
detects Carmichael numbers.
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Idea: a(n-1)/2 = {1,-1}, how about a(n-1)/4? If a(n-1)/4 = {1,-1},
how about a(n-1)/8?
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Prove by contradiction.
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a2B,
Assume n is Carmichael, for all a, aB = 1 mod n.
Property: Carmichael number is the product of distinct prime. Thus,
let n = p1p2..pk.
Let g’ is a generator of Zp1*.
Let a = (g’, 1), i.e., a = g’ (mod p1), a = 1 (mod p2..pr), by CRT
By assumption, aB = 1 (mod n). It implies g’B = 1 (mod p1) (why?).
Since g’ is the generator, B = p-1, which contradicts B is odd.
Thus, for some a, aB <> 1. The probability is > ½.
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How to Find a Prime Number?
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Algorithm:
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Randomly pick a number from [1,n-1].
Plug it into the primality testing algorithm.
If fails, repeat the test with another number.
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Are prime numbers rare? No.
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Prime number theorem:
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No. of prime numbers less than n ~ n/ln(n).
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References
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R. Motwani and P. Raghavan, “Randomized
Algorithms”, Ch. 14.
CMU, “Randomized algorithms”, http://www2.cs.cmu.edu/afs/cs/usr/avrim/www/Randalg
s98/home.html
CLRS, “Introduction to Algorithms”, 2nd
edition. Ch. 31.
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