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 ∈ 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 | a and n | b it holds that a = nk, b = np for some k, p ∈ Z. From a = bq + r we get r = bq − a expanding to r = nk − npq = n(k − pq) and since k − pq ∈ Z then n | 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 ∈ 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 | a and n | b it holds that a = nk, b = np for some k, p ∈ Z. From a = bq + r we get r = bq − a expanding to r = nk − npq = n(k − pq) and since k − pq ∈ Z then n | 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 | b ∧ m | r → b = km, r = pm, k, p ∈ Z

a = bq + r = km + mp = m(kq + p) kq + p ∈ Z So m | 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 | (a − b)

Euclidean Algorithm and Congruence Let a, b, c, d, k ∈ Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k).

From the definition of mod we get k | a − b and k | c − d.

From linearity of divides we get k | (a − b) + (c − d) and then k | (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 | (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 | (a − b)

Let a, b, c, d, k ∈ Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k).

From the definition of mod we get k | a − b and k | c − d.

From linearity of divides we get k | (a − b) + (c − d) and then k | (a + b) − (b + d) so (a + c) ≡ (b + d) (mod k). QED

Euclidean Algorithm and Congruence Let a, b, c, d, k ∈ Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k). From the definition of mod we get k | a − b and k | 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 | (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 | (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 | (a − b)

Let a, b, c, d, k ∈ Z with k > 0 s.t. a ≡ b (mod k) and c = d (mod k). From the definition of mod we get k | a − b and k | 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 | (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] = {..., −11, −4, 3, 10, 17,... }

In ( mod 5), [3] = {..., −7, −2, 3, 8, 13,... }

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