Euclidean Algorithm Congruence Euclidean Algorithm and Congruence G. Carl Evans University of Illinois Fall 2018 Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Recall Euclidean Algorithm Remainder(a; b) is the remainder when a is divided by b. RecursiveGCD (a; b) r Remainder(a; b) if r = 0 return b return RecursiveGCD (b; r) Euclidean Algorithm and Congruence Euclidean Algorithm Congruence But why does Euclidean algorithm work? RecursiveGCD (a; b) r Remainder(a; b) if r = 0 return b return RecursiveGCD (b; r) The Euclidean algorithm works iff gcd(a; b) = gcd(b; r), where r = the remainder of the remainder when a is divided by b. Euclidean Algorithm and Congruence Let a; b; q; r 2 Z with b > 0 and a = bq + r. Let n = gcd(a; b) and m = gcd(b; r) So n is the largest integer that divides both a and b and m is the largest integer that divides both b and r. Since n j a and n j b it holds that a = nk; b = np for some k; p 2 Z. From a = bq + r we get r = bq − a expanding to r = nk − npq = n(k − pq) and since k − pq 2 Z then n j r. Euclidean Algorithm Congruence Proof of Euclidean algorithm Claim: For any integers a; b; q; r; with b > 0, if a = bq + r then gcd(a; b) = gcd(b; r). Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Proof of Euclidean algorithm Claim: For any integers a; b; q; r; with b > 0, if a = bq + r then gcd(a; b) = gcd(b; r). Let a; b; q; r 2 Z with b > 0 and a = bq + r. Let n = gcd(a; b) and m = gcd(b; r) So n is the largest integer that divides both a and b and m is the largest integer that divides both b and r. Since n j a and n j b it holds that a = nk; b = np for some k; p 2 Z. From a = bq + r we get r = bq − a expanding to r = nk − npq = n(k − pq) and since k − pq 2 Z then n j r. Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Proof of Euclidean algorithm pt2 Claim: For any integers a; b; q; r; with b > 0, if a = bq + r then gcd(a; b) = gcd(b; r). Similarly: m j b ^ m j r ! b = km; r = pm; k; p 2 Z a = bq + r = km + mp = m(kq + p) kq + p 2 Z So m j a. Since n and m divide a; b and r there are three cases either n < m; m < n or; n = m. Since n is the largest value that divides a and b, n is not less then m. Similarly since m is the largest value that divides b and r, m is not less then n. Thus m = n. QED Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Applications of congruence bitwise operations error checking encryption telling time etc. Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Congruence mod k Two integers are congruent mod k if the differ by an integer multiple of k Definition: If k is any positive integer, two integers a and b are congruent mod k if k divides (a − b). a ≡ b( mod k) $ k j (a − b) Euclidean Algorithm and Congruence Let a; b; c; d; k 2 Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k). From the definition of mod we get k j a − b and k j c − d. From linearity of divides we get k j (a − b) + (c − d) and then k j (a + b) − (b + d) so (a + c) ≡ (b + d) (mod k). QED Euclidean Algorithm Congruence Modulus addition proof Claim: For any integers a; b; c; d; k with k > 0, if a ≡ b (mod k) and c ≡ d (mod k) then (a + c) ≡ (b + d) (mod k). Definition: a ≡ b (mod k) $ k j (a − b) Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Modulus addition proof Claim: For any integers a; b; c; d; k with k > 0, if a ≡ b (mod k) and c ≡ d (mod k) then (a + c) ≡ (b + d) (mod k). Definition: a ≡ b (mod k) $ k j (a − b) Let a; b; c; d; k 2 Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k). From the definition of mod we get k j a − b and k j c − d. From linearity of divides we get k j (a − b) + (c − d) and then k j (a + b) − (b + d) so (a + c) ≡ (b + d) (mod k). QED Euclidean Algorithm and Congruence Let a; b; c; d; k 2 Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k). From the definition of mod we get k j a − b and k j c − d. So (a − b) = nk and (c − d) = pk by def divides. This leads to a = nk + b and c = pk + d and ac = (nk+b)(pk+d) = pnk2+dnk+bpk+bd = bd+(pnk+dn+b)k Since pnk + dn + b is an integer ac = bd + qk where q is an integer. Thus ac − bd = qk and k j (ac − bd) and ac ≡ bd (mod k). QED Euclidean Algorithm Congruence Modulus multiplication proof Claim: For any integers a; b; c; d; k with k > 0, if a ≡ b (mod k) and c ≡ d (mod k) then ac ≡ bd( mod k). Definition: a ≡ b( mod k) $ k j (a − b) Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Modulus multiplication proof Claim: For any integers a; b; c; d; k with k > 0, if a ≡ b (mod k) and c ≡ d (mod k) then ac ≡ bd( mod k). Definition: a ≡ b( mod k) $ k j (a − b) Let a; b; c; d; k 2 Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k). From the definition of mod we get k j a − b and k j c − d. So (a − b) = nk and (c − d) = pk by def divides. This leads to a = nk + b and c = pk + d and ac = (nk+b)(pk+d) = pnk2+dnk+bpk+bd = bd+(pnk+dn+b)k Since pnk + dn + b is an integer ac = bd + qk where q is an integer. Thus ac − bd = qk and k j (ac − bd) and ac ≡ bd (mod k). QED Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Equivalence classes with modulus The equivalence class of integer x (written [x]) is the set of all integers congruent to x( mod k). In ( mod 7); [3] = f:::; −11; −4; 3; 10; 17;::: g In ( mod 5); [3] = f:::; −7; −2; 3; 8; 13;::: g In Z5; [3] = [8] = [−2] Euclidean Algorithm and Congruence Euclidean Algorithm Congruence Modulus arithmetic [x] + [y] = [x + y] [x] ∗ [y] = [x ∗ y] Euclidean Algorithm and Congruence.