1. Numerical Algorithms x 1 2 3 4 5 6 7 8 9 x-1 Numerical Algorithms
6/20/2018 9:29 AM
Numerical Algorithms x 1 2 3 4 5 6 7 8 9
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Numerical Algorithms

2. Outline Divisibility and primes Modular arithmetic
Euclid’s GCD algorithm
Multiplicative inverses
Powers
Fermat’s little theorem
Euler’s theorem
Numerical Algorithms

3. Divisibility (1)
Definition If a and b are integers with a ≠ 0, we say that a divides b if there is an integer c such that b = a∙c.
When a divides b we say that a is a factor of b and that b is a multiple of a.
The notation a|b denotes that a divides b.
Numerical Algorithms

4. Divisibility (2) Theorem Let a, b, and c be integers (a, b, c  Z).
If a|b and a|c, then a|(b+c).
If a|b, then a|b∙c for all integers c.
If a|b and b|c, then a|c.
Corollary If a, b, and c are integers (a, b, c  Z) such that a|b, and a|c, then a|m∙b+n∙c, for every pair of integers m and n.
Theorem If a and b are positive integers and a|b, then a ≤ b.
Note: 1|n for any integer n, and n|0 for any integer n ≠0.
Numerical Algorithms

5. Fundamental Theorem of Arithmetic
Prime number p:
p is an integer
p  2
The only divisors of p are 1 and p
Fundamental Theorem of Arithmetic
The prime decomposition n = p1e1  …  pkek
of a positive integer n is unique
Examples:
200 = 23  52
100 = 2 ∙ 2 ∙ 5 ∙ 5 = 22 ∙ 52
641 = 641
999 = 3 ∙ 3 ∙ 3∙ 37 = 33 ∙ 37
1024 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 210
Numerical Algorithms

6. Greatest Common Divisor
The greatest common divisor (GCD) of two positive integers a and b, denoted gcd(a, b), is the largest positive integer that divides both a and b
The above definition is extended to arbitrary integers
Examples:
gcd(18, 30) = 6 gcd(0, 20) = 20 gcd(-21, 49) = 7
Two integers a and b are said to be relatively prime if
gcd(a, b) = 1
Example:
Integers 15 and 28 are relatively prime
Numerical Algorithms

7. Modular Arithmetic gcd(a, b) = gcd(b, a mod b)
Modulo operator for a positive integer n
r = a mod n
equivalent to
a = r + kn
and
r = a - a/n n
Example:
29 mod 13 = 3 13 mod 13 = 0 -1 mod 13 = 12
29 = 3 + 213 13 = 0 + 113 12 = 13
Modulo and GCD:
gcd(a, b) = gcd(b, a mod b)
gcd(21, 12) = 3 gcd(12, 21 mod 12) = gcd(12, 9) = 3
Numerical Algorithms

8. Euclid’s GCD Algorithm
Euclid’s algorithm for computing the GCD repeatedly applies the formula
gcd(a, b) = gcd(b, a mod b)
Example
gcd(412, 260) = 4
Algorithm EuclidGCD(a, b)
Input integers a and b
Output gcd(a, b)
if b = 0
return a
else
return EuclidGCD(b, a mod b) a
412
260
152
108 44 20 4 b
Numerical Algorithms

9. Analysis
Let ai and bi be the arguments of the i-th recursive call of algorithm EuclidGCD
We have
ai + 2 = bi + 1 = ai mod ai + 1 < ai + 1
Sequence a1, a2, …, an decreases exponentially, namely
ai + 2  ½ ai for i > 1
Case 1 ai + 1  ½ ai ai + 2 < ai + 1  ½ ai
Case 2 ai + 1 > ½ ai ai + 2 = ai mod ai + 1 = ai - ai + 1  ½ ai
Thus, the maximum number of recursive calls of algorithm EuclidGCD(a. b) is
1 + 2 log max(a. b)
Algorithm EuclidGCD(a, b) executes O(log max(a, b)) arithmetic operations
Numerical Algorithms

10. Congruencies (1)
Definition For integers a and n the expression a mod n is the reminder r of a when divided by n. In other words there is some integer q such that a = qn + r.
Definition If a mod n = b mod n we say that a is congruent to b modulo n and we write
a ≡ b mod n
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11. Congruencies (2) Theorem (a) If a ≡ x (mod n) and b ≡ y (mod n), then
a + b ≡ x + y (mod n), and a∙b ≡ x∙y (mod n).
(b) If a∙c ≡ b∙c (mod n) and gcd(c, n) = 1, then
a ≡ b (mod n).
(c) If a ≡ b (mod n), then am ≡ bm (mod n) for all
positive integers m.
(d) If a ≡ b (mod m∙n), then a ≡ b (mod m) and
(f) If a ≡ b (mod n), a ≡ b (mod m), and gcd(m, n) = 1, then a ≡ b (mod m∙n).
Numerical Algorithms

12. Additive inverses The set of residues modulo a positive integer n is
Zn = {0, 1, 2, …, (n - 1)}
Theorem Every element x in Zn has an additive inverse modulo n, i.e. there is a y in Zn such that (x + y) mod n = 0.
Example: The additive inverse of 4 modulo 9 is 5.
Numerical Algorithms

13. Multiplicative Inverses (1)
Let x and y be two elements of Zn such that
xy mod n = 1
We say that y is the multiplicative inverse of x in Zn and we write y = x-1
Example:
Multiplicative inverses of the residues modulo 11 x 1 2 3 4 5 6 7 8 9 10
x-1
Numerical Algorithms

14. Multiplicative Inverses (2)
Theorem
An element x of Zn has a multiplicative inverse if and only if x and n are relatively prime
Example
The elements of Z10 with a multiplicative inverse are 1, 3, 7, 9
Corollary
If p is prime, every nonzero residue in Zp has a multiplicative inverse
A variation of Euclid’s GCD algorithm computes the multiplicative inverse of an element x of Zn or determines that it does not exist x 1 2 3 4 5 6 7 8 9
x-1
Numerical Algorithms

15. Powers
Let p be a prime
The sequences of successive powers of the elements of Zp exhibit repeating subsequences
The sizes of the repeating subsequences and the number of their repetitions are the divisors of p - 1
Example (p = 7) x x2 x3 x4 x5 x6 1 2 4 3 6 5
Numerical Algorithms

16. Fermat’s Little Theorem
Let p be a prime. For each nonzero residue x of Zp, we have xp - 1 mod p = 1
Example (p = 5):
14 mod 5 = mod 5 = 16 mod 5 = 1
34 mod 5 = 81 mod 5 = mod 5 = 256 mod 5 = 1
Corollary
Let p be a prime. For each nonzero residue x of Zp, the multiplicative inverse of x is xp - 2 mod p
Proof
x(xp - 2 mod p) mod p = xxp - 2 mod p = xp - 1 mod p = 1
Numerical Algorithms

17. Euler’s Theorem (1)
The multiplicative group for Zn, denoted with Z*n, is the subset of elements of Zn relatively prime with n
The totient function of n, denoted with f(n), is the size of Z*n
Example
Z*10 = { 1, 3, 7, 9 } f(10) = 4
If p is prime, we have
Z*p = {1, 2, …, (p - 1)} f(p) = p - 1
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18. Euler’s Theorem (2) Euler’s Theorem
For each element x of Z*n, we have xf(n) mod n = 1
Example (n = 10)
3f(10) mod 10 = 34 mod 10 = 81 mod 10 = 1
7f(10) mod 10 = 74 mod 10 = 2401 mod 10 = 1
9f(10) mod 10 = 94 mod 10 = 6561 mod 10 = 1
Numerical Algorithms

19. Modular Exponentiation (1)
The repeated squaring algorithm speeds up the computation of a modular power ap mod n
Write the exponent p in binary
p = pb - 1 pb - 2 … p1 p0
Start with
Q1 = apb - 1 mod n
Repeatedly compute
Qi = ((Qi - 1)2 mod n)apb - i mod n
We obtain
Qb = ap mod n
The repeated squaring algorithm performs O (log p) arithmetic operations
Numerical Algorithms

20. Modular Exponentiation (2)1
3p5 - i 3 Qi 9 5 18
Example
318 mod 19 (18 = 10010)
Q1 = 31 mod 19 = 3
Q2 = (32 mod 19)30 mod 19 = 9
Q3 = (92 mod 19)30 mod 19 = 81 mod 19 = 5
Q4 = (52 mod 19)31 mod 19 = (25 mod 19)3 mod 19 = = 18 mod 19 = 18
Q5 = (182 mod 19)30 mod 19 = (324 mod 19) mod 19 = 17 mod 19 = 1
Numerical Algorithms

21. Modular Multiplicative Inverses (1)
Theorem
Given positive integers a and b, let d be the smallest positive integer such that
d = ia + jb
for some integers i and j.
We have
d = gcd(a,b)
Example
a = 21
b = 15
d = 3
i = 3, j = -4
3 = 321 + (-4)15 = = 3
Numerical Algorithms

22. Modular Multiplicative Inverses (2)
Algorithm ExtendedEuclidGCD(a, b)
Input: Nonnegative integers a and b
Output: Triplet of integers (d, i, j) such that d=gcd(a, b)=ia+jb
if b = 0 then
return (a, 1, 0)
q← a mod b
Let r be the integer such that a = rb+q
(d,k,l) ← ExtendedEuclidGCD(b,q)
return (d, l, k-lr)
Numerical Algorithms

23. Modular Multiplicative Inverses (3)
Theorem Given positive integers a and b, the extended Euclid’s algorithm for computing a triplet (d, i, j) such that
d = gcd(a,b)
d = ia + jb
executes in O(log max(a, b)) arithmetic operations.
Corollary Let x be an element of Zn such that gcd(x, n) = 1. The multiplicative inverse of x in Zn can be computed with O(log n) arithmetic operations.
Numerical Algorithms

24. Modular Multiplicative Inverses (4)
To test the existence of and compute the inverse of x  Zn, execute the extended Euclid’s algorithm on the input pair (x,n)
Let (d,i,j) be the triplet returned
d = ix + jn
Case 1: d = 1
i is the inverse of x in Zn
Case 2: d > 1
x has no inverse in Zn
Numerical Algorithms

25. Pseudoprimality Testing (1)
The number of primes less than or equal to n is about n / ln n
Thus, we expect to find a prime among, O(b) randomly generated numbers with b bits each
Testing whether a number is prime (primality testing) is believed to be a hard problem
An integer n  2 is said to be a base-x pseudoprime if
xn - 1 mod n = 1 (Fermat’s little theorem)
Numerical Algorithms

26. Pseudoprimality Testing (2)
Composite base-x pseudoprimes are rare:
A random 100-bit integer is a composite base-2 pseudoprime with probability less than 10-13
The smallest composite base-2 pseudoprime is 341
Base-x pseudoprimality testing for an integer n:
Check whether xn - 1 mod n = 1
Can be performed efficiently with the repeated squaring algorithm
Numerical Algorithms

27. Randomized Primality Testing
Compositeness witness function witness(x, n) with error probability q for a random variable x
Case 1: n is prime
witness w(x, n) = false
Case 2: n is composite
witness w(x, n) = false with probability q < 1
Algorithm RandPrimeTest tests whether n is prime by repeatedly evaluating witness(x, n)
A variation of base- x pseudoprimality provides a suitable compositeness witness function for randomized primality testing (Rabin-Miller algorithm)
Algorithm RandPrimeTest(n, k)
Input integer n,confidence parameter k and composite witness function witness(x,n) with error probability q
Output an indication of whether n is composite or prime with probability 2-k
t  k/log2(1/q)
for i  1 to t
x  random()
if witness(x,n)= true
return “n is composite”
return “n is prime”
Numerical Algorithms