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Greatest Common Divisor

Greatest Common Divisor. We now know that if Z n is a field, then n must be a prime number. Is it true that Z p is a field for all prime p? We will answer the more general question of which elements of Z n have multiplicative inverses (useful for affine ciphers)

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Greatest Common Divisor

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  1. Greatest Common Divisor • We now know that if Zn is a field, then n must be a prime number. • Is it true that Zp is a field for all prime p? • We will answer the more general question of which elements of Zn have multiplicative inverses (useful for affine ciphers) • DefinitionIf m and n are integers, not both zero, the greatest common divisor of m and n, denoted gcd(m,n), is the largest integer that divides both m and n • gcd(0,0) is not defined, since every integer divides 0 • Symbolically: gcd(m,n) = max { k : k | m and k | n } • Example: gcd(24,32) = 8 • Since 1 is a divisor of any integer, gcd(m,n)  1 • If m  0, then gcd(m,n) ≤ |m|

  2. Greatest Common Divisor • We now state some elementary properties of the gcd function 1. For integers m and n, not both zero, gcd(m,n) = gcd(|m|,|n|). 2. If m> 0, then gcd(m,m) = m and gcd(m,0) = m 3. For integers m and n, not both zero, gcd(m,n) = gcd(n,m). 4. For integers m and n, not both zero, gcd(m,n) = gcd(m+kn,n) for any integer k

  3. Euclid’s Greatest Common Divisor Algorithm • Since m mod n = m - m/nn, We can now apply statement 4 and statement 3 to see that gcd(m,n) = gcd(m mod n, n) = gcd(n, m mod n) 5. For integers m and n with n > 0, gcd(m,n) = gcd(n, m mod n). • We can apply statement 5 to derive an algorithm for computing - greatest common divisors • Euclid’s Algorithm: • Input: integers m and n, not both zeroOutput: d = gcd(m,n) • If n == 0 d = |m|elsewhile n  0 c = n n = m mod n m = c d = mreturn d

  4. Euclid’s Greatest Common Divisor Algorithm • Example 1: compute gcd(24,32) • gcd(24,32) = gcd(32,24) = gcd(24,32 mod 24) = gcd(24,8) = gcd(8, 24 mod 8) = gcd(8,0) = 8

  5. Euclid’s Greatest Common Divisor Algorithm • One of the special properties of the greatest common divisor of two numbers is that it can be written as an integer linear combination of the numbers • Example 2: gcd(32,24) = 8 • 32 = 124 + 8 • Thus 8 = 132 + (-1)24 • Example 3: gcd(54,42) 54 = 142 + 12 42 = 312 + 6 12 = 26 + 0, so 6 = gcd(54,42) Using back-substitution: 6 = 142 - 312 and 12 = 154 - 142 Thus 6 = 142 - 3(154 - 142 ) = 442 + (-3)54

  6. Greatest Common as a Linear Combination TheoremLet a and b be integers, not both zero. Then there are integers x and y such that gcd(a,b) = xa + yb. • Why is this result so interesting to us? • Suppose we are looking at an element a of Zn for some n > 1. • If gcd(a,n) = 1, then there are integers x, y such that 1 = xa + yn • Apply the Division Algorithm to find the quotient q and remainder x mod n when dividing x by n • Since x = qn + (x mod n), we have xa = (x mod n)a + qna • Thus 1 = xa + yn = ((x mod n)a + qna) + yn = (x mod n)a + (qa + y)n • Therefore (x mod n)a = 1 – (qa+y)n = 1 (mod n) • Since x mod n < n, we have found a multiplicative inverse for a in Zn.

  7. Greatest Common as a Linear Combination • Suppose p is prime • Since every element in {1,…p-1} is relativel prime to p, we see that there is another element b in Zp such that an b =1. That is, every nonzero element of Zp has a multiplicative inverse in Zp, which means that: • If p is prime, then (Zn , n , n ) is a field. • More importantly, in an affine cipher y = ax + b, we must choose a so that gcd(a,26) = 1.

  8. Extended Euclidean Algorithm • We now want an algorithm that computes not only the gcd(a,b) but also the coefficents x and y so that gcd(a,b) = xa + ybn • This would enable us to compute multiplicative inverses in Zn for general n. • This would be particularly useful for computing the decryption function for an affine cipher • If the affine cipher encryption function is given by y = ax + b, then the decryption function would be x = a-1(y+(-b)), where –b is the additive inverse of b and a-1 is the multiplicative inverse of a in Z26. • Example: Since gcd(9,26) = 1, 9 has an inverse in Z26. • Thus y = 9x + 4 is a valid affine encryption function. • It is easy to see that the multiplicative inverse of 9 in Z26 is 3: 39 = 27 and 27 mod 26 = 1, so 3 26 9 = 1 • Also, additive inverse of 4 in Z26 is 22, since 22 + 4 = 26 and 26 mod 26 = 0 • Thus the corresponding decryption function is x = 3(y+(-4)) = 3(y + 22) = 3y + 3 n 22 = 3y + (66 mod 26) = 3y + 14

  9. Extended Euclidean Algorithm • The idea is to compute a sequence of “remainders” r0, r1, r2, … terminating with the greatest common divisor of the two numbers a and b. • We also want to compute, as we go, coefficients xi and yi such that ri = xia + yib • Actually, r2 will be our first true remainder: we start with r0= a and r1 = b. • It is then obvious what the coefficients should be: x0 = 1, y0 = 0; x1 = 0, y1 = 1 • Why? Because a = 1a + 0b and b = 0a + 1b • Now we set r2 = r0 mod r1 • How do we get the coefficients xi and yi ? From the integer division theorem: • r0 =  r0/ r1r1 + (r0 mod r1) =  r0/ r1r1 + r2 • Thus r2 = r0 -  r0/ r1r1 = (x0a + y0b) -  r0/ r1(x1a + y1b) = (x0-  r0/ r1x1)a + (y0-  r0/ r1y1)b • So we set x2 = x0-  r0/ r1x1 and y2 = y0-  r0/ r1y1 • In general, ri+1 = ri-1 mod ri, xi+1 = xi-1-  ri-1/ rixi, yi+1 = yi-1-  ri-1/ riyi • Notice that when we get rn+1 = 0, then rn = gcd(a,b)

  10. Extended Euclidean Algorithm • So here is the algorithm: r0= a , r1 = b, x0 = 1, y0 = 0; x1 = 0, y1 = 1 (initialization) i = 1 Do the following while ri 0: • ri+1 = ri-1 mod ri, xi+1 = xi-1-  ri-1/ rixi, yi+1 = yi-1-  ri-1/ riyi • i = i+1 return ri-1

  11. Extended Euclidean Algorithm A computationally more appealing method can be derived from the following observation:

  12. Extended Euclidean Algorithm Thus we have the following matrix version of the algorithm: Input: Integers a and b, not both zero Output: Integers x,y and d such that d = gcd(a,b) and d = xa + yb while d1 0 do: d = d0, x = x0, y = y0 return d, x, y

  13. Example 4: a = 52, b = 96 gcd(52,96) = 4 4 = (-11) 52 + 6 96

  14. Relatively Prime Pairs • DefinitionIf gcd(a,b) = 1, then we say that a and b are relatively prime and write a  b • TheoremIf a | bc and a  b, then a | c • Corollary If p is a prime and p | ab then p | a or p | b. • Restating an earlier result: • Integer a  Zn has a n-inverse in Zn if and only if a  n.

  15. Algorithm for Computing b-inverses Recall that if gcd(a,b) = 1, then a has a multiplicative inverse in Zb Moreover, we showed that if 1 = xa + yb, then x mod b is the b-inverse of a Thus, we do not need to compute the y in the extended Euclidean algorithm In fact, we can eliminate the last column of the 2 by 3 matrix in that algorithm. Input: Integers a and b with b > 1. Output: The multiplicative inverse of a mod b. while d1 0 do: if d0  1 return “No inverse” else return x0

  16. Example: Computing b-inverses 4. Compute the multiplicative inverse of 43 in Z56 Thus the 56-inverse of 43 is x = (-13 mod 56) = 56+(-13) = 43

  17. Least Common Multiple LemmaIf a = and b = , where the pts are distinct primes andeach exponent is nonnegative, then gcd(a,b) = DefinitionThe least common multiple of positive integers a and b is the least integer divisible by both a and b. Notation: lcm(a,b) LemmaIf a = and b = , where the pts are distinct primes and each exponent is nonnegative, then lcm(a,b) = TheoremIf a and b are positive integers, then lcm(a,b) =

  18. Modular Arithmetic • DefinitionGiven integers a, b and m > 0, we say a is congruent to b mod m, written a  b (mod m) or a m b, if and only if a mod m = b mod m. • TheoremIf a, b, m are integers with m > 0, then a m b if and only if m | (b-a) • DefinitionA set C of integers is a complete residue system modulo m iff •  integer a  c  C such that a m c •  c, d  C, if c m d then c = d • Thus C is a complete residue system modulo m if and only if every integer is congruent mod m to exactly one element of C. • The set { 0, 1, . . . , m-1 } of remainders mod m is the least nonnegative complete residue system mod m.

  19. Basic Properties of the Congruence Relation TheoremLet m and d be positive integers and a, b, c arbitrary integers. Then (i) a m a (ii) a m b b m a (iii) a m b and b m c a m c (iv) a m b a +c m b +c and ac m bc (v) a m b ad m bd (vi) a m and ab m ac  b m c (vii) gcd(a,b) = d  a/d  b/d (viii) gcd(a,m) = d and ab m ac  bd m cd Note: statement (viii) in the book is false, as is the last statement of the proof! (Let m = 6, a = 2, b = 3 and c = 6).

  20. Homework • Find d = gcd(43,56) and integers x and y such that d = 43x + 56y, using the extended Euclidean algorithm • For this problem, suppose we are using an affine cipher with encryption function y = 9x + 4 • Encrypt the message “nothard”; your answer should be a string of capital letters • Decrypt the message “TAAOEKM”; your answer should be a string of lower case letters

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