ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse...
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Modular Arithmetic
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 2
Previously, we defined
Def (Divisibility). We say that a divides b if there is an integer ksuch that
b = a · k.
We write a | b if a divides b. Otherwise, we write a - b.
Theorem (The Division Algorithm). Let a be an integer and d apositive integer. Then there are unique integers q and r, such that0≤ r < d and
a = dq+ r.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 3
Fundamental theorem of arithmetic
Def (Prime numbers). A number p > 1 with no positive divisorsother than 1 and itself is called a prime.
Every other number greater than 1 is called composite.
The number 1 is considered neither prime nor composite.
Theorem (Fundamental theorem of arithmetic). Every positiveinteger n can be written in a unique way as a product of primes
n= p1 · p2 · . . . · p j (p1 ≤ p2 ≤ . . .≤ p j)
This product is called prime factorization.
See Lehman and Leighton (p. 67) for the proof.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 4
Divisibility by a prime
One more result about primes we are going to use in the future:
Theorem. Let p be a prime. If
p | a1a2 · . . . · an,
then p divides some ai (at least one of them).
Example: If you know that 19 | 403·629, then you know that either19 | 403 or 19 | 629, though you might not know which.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 5
GCD is a linear combination
Def (GCD).Theorem (Bezout’s Theorem). If a and b are positive integers,then there exist integers s and t such that
gcd(a, b) = sa+ t b.
Exmaple: gcd(52,44) = 4
6 · 52+ (−7) · 44= 4
So called Extended Euclid’s algorithm constructs such s and t, andso proves the theorem. The algorithm is described in the last sec-tion of this lecture.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 6
Relative primes (co-primes)
Def (Relative primes). a and b are relative primes (or co-primes) if
gcd(a, b) = 1.
By Bezout’s theorem, a and b are co-primes if and only if there exists and t such that
sa+ t b = 1
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 7
Modular arithmetic
. . .− 2 − 1 → 0 1 2 3 4 5 6 → 7 8 . . .
What if instead of integers, we deal with
a finite set of periodically repeating integers?
. . . 5 6 → 0 1 2 3 4 5 6 → 0 1 . . .
For example, the days of the week behave in this way.
Mon, Tue, Wed, Thr, Fri, Sat, Sun,
are followed again by Mon, Tue, and so on.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 8
Modular arithmetic
. . .− 2 − 1 → 0 1 2 3 4 5 6 → 7 8 . . .
What if instead of integers, we deal with
a finite set of periodically repeating integers?
. . . 5 6 → 0 1 2 3 4 5 6 → 0 1 . . .
For example, the days of the week behave in this way.
Sun, Mon, Tue, Wed, Thr, Fri, Sat,
are followed again by Sun, Mon, and so on.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 9
Modular arithmetic
. . . 5 6 → 0 1 2 3 4 5 6 → 0 1 . . .
We want to add, subtract, multiply, and, hopefully, divide such spe-cial “integers” . . .
4+ 4 is 1
2− 3 is 6
14 · 5 is 0
−7 is 0 is 7 is 14 is 21 . . .
First, we need to rigorously define, which integers can be called“equal” in such modular arithmetic. We will call them congruent.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 10
CongruenceDef. For a positive integer n, a is congruent to b modulo n if
n | (a− b).
This is denoteda ≡ b (mod n).
Example:
8≡ 1 (mod 7)15≡ 1 (mod 7)8≡ 15 (mod 7)
because7 | (8− 1︸︷︷︸
=7
), 7 | (15− 1︸ ︷︷ ︸
=14
), 7 | (15− 8︸ ︷︷ ︸
=7
).
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 11
CongruenceLemma. If a ≡ b (mod n), then exists k ∈ Z s.t. a = b+ kn.
Lemma. Two numbers are congruent modulo n if and only if theyhave the same remainder when divided by n.
a ≡ b (mod n) if and only if a rem n= b rem n.
Proof: By the division algorithm,
a = q1n+ r1, b = q2n+ r2.
a− b = (q1 − q2)n+ (r1 − r2)
“⇒”: If a ≡ b (mod n) then n | (a − b). So r1 − r2 = 0, theremainders are equal.
“⇐”: If r1 = r2, then n | (a− b), so a ≡ b (mod n).
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 12
Congruence-3 0 3 6
-2 1 4 7-1 2 5 8
x rem 3 0 1 2 0 1 2 0 1 2 0 1 2
Integers are divided into 3 congruence classes:
. . . , -3, 0, 3, 6, 9, 12, . . . are congruent modulo 3.
. . . , -2, 1, 4, 7, 10, 13, . . . are congruent modulo 3.
. . . , -1, 2, 5, 8, 11, 14, . . . are congruent modulo 3.
By the way, all Mondays, all Tuesdays, all Wednesdays, etc. arecongruence classes too.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 13
Modular arithmeticAddition, subtraction, and multiplication preserve congruence.
Theorem. if a ≡ b (mod n) and c ≡ d (mod n), then
a+ c ≡ b+ d (mod n).
Theorem. if a ≡ b (mod n) and c ≡ d (mod n), then
ac ≡ bd (mod n).
Proof.Exist x , y ∈ Z such that a− b = xn and c − d = yn.
ac − bd = (b+ xn)(d + yn)− bd = n(xd + b y + xny)
Thus ac ≡ bd (mod n).
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 14
Modular arithmeticAddition, subtraction, and multiplication preserve congruence.What does it mean practically?
If we have to find x such that
122 · (−11) + 80≡ x (mod 5)
We know that
12≡ 2 (mod 5),−11≡ −1 (mod 5),
80≡ 0 (mod 5)
Therefore, we are free to substitute 12 with 2, −11 with −1, and80 with 0:
122 ·(−11)+80 ≡ 22 ·(−1)+0 ≡ 2 ·2 ·(−1) ≡ −4 ≡ 1 (mod 5).
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 15
Multiplicative inverse
What about division?
Theorem. if a and n are relative primes, i.e. gcd(a, n) = 1, thenexists integer a−1 called multiplicative inverse, such that
aa−1 ≡ 1 (mod n)
Proof.Exist s and t, such that sa+ tn= 1. Therefore,
sa− 1= tn
sa ≡ 1 (mod n)
Therefore, a−1 = s.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 16
Multiplicative inverse
Corollary. If a and n are relative primes, then there exists a uniquemultiplicative inverse a−1 ∈ {1, 2, . . . , n− 1} such that
aa−1 ≡ 1 (mod n).
Ok, uniqueness is great, but we need a procedure for finding mul-tiplicative inverses.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 17
Multiplicative inverse
Find inverse of 101 modulo 4620, that is x such that
101 · x ≡ 1 (mod 4620)
If 101 and 4620 are relative primes:
gcd(101, 4620) = 1,
by Bezout’s theorem: Exist s and t such that
101 · s+ 4620 · t = gcd(101, 4620) = 1
101 · s ≡ 1 (mod 4620)
We have to find Bezout coefficients s and t. Then s is the inverse.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 18
Multiplicative inverse
Find inverse of 101 modulo 4620, that is x such that
101 · x ≡ 1 (mod 4620)
If 101 and 4620 are relative primes:
gcd(101, 4620) = 1,
by Bezout’s theorem: Exist s and t such that
101 · s+ 4620 · t = gcd(101, 4620) = 1
101 · s ≡ 1 (mod 4620)
We have to find Bezout coefficients s and t. Then s is the inverse.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 19
Recall Euclid’s Algorithm
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Let’s adapt this algorithm for finding Bezout coefficients s and t:
101 · s+ 4620 · t = 1
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 20
Extended Euclid’s Algorithm
101 · s+ 4620 · t = 1
Run Euclid’s algorithm:
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Work backwards, to express GCD interms of a1 = 101 and a0 = 4620:
1= 3− 1 · 2= 3− 1 · (23− 7 · 3) = −1 · 23+ 8 · 3= −1 · 23+ 8 · (26− 1 · 23) = 8 · 26− 9 · 23= 8 · 26− 9(75− 2 · 26) = −9 · 75+ 26 · 26= −9 · 75+ 26 · (101− 1 · 75) = 26 · 101− 35 · 75= 26 · 101− 35 · (4620− 45 · 101)
= −35 · 4620+ 1601 · 101
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 21
Extended Euclid’s Algorithm
101 · s+ 4620 · t = 1
Run Euclid’s algorithm:
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Work backwards, to express GCD interms of a1 = 101 and a0 = 4620:
1= 3− 1 · 2= 3− 1 · (23− 7 · 3) = −1 · 23+ 8 · 3= −1 · 23+ 8 · (26− 1 · 23) = 8 · 26− 9 · 23= 8 · 26− 9(75− 2 · 26) = −9 · 75+ 26 · 26= −9 · 75+ 26 · (101− 1 · 75) = 26 · 101− 35 · 75= 26 · 101− 35 · (4620− 45 · 101)
= −35 · 4620+ 1601 · 101
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 22
Extended Euclid’s Algorithm
101 · s+ 4620 · t = 1
Run Euclid’s algorithm:
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Work backwards, to express GCD interms of a1 = 101 and a0 = 4620:
1= 3− 1 · 2= 3− 1 · (23− 7 · 3) = −1 · 23+ 8 · 3= −1 · 23+ 8 · (26− 1 · 23) = 8 · 26− 9 · 23= 8 · 26− 9(75− 2 · 26) = −9 · 75+ 26 · 26= −9 · 75+ 26 · (101− 1 · 75) = 26 · 101− 35 · 75= 26 · 101− 35 · (4620− 45 · 101)
= −35 · 4620+ 1601 · 101
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 23
Extended Euclid’s Algorithm
101 · s+ 4620 · t = 1
Run Euclid’s algorithm:
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Work backwards, to express GCD interms of a1 = 101 and a0 = 4620:
1= 3− 1 · 2= 3− 1 · (23− 7 · 3) = −1 · 23+ 8 · 3= −1 · 23+ 8 · (26− 1 · 23) = 8 · 26− 9 · 23= 8 · 26− 9(75− 2 · 26) = −9 · 75+ 26 · 26= −9 · 75+ 26 · (101− 1 · 75) = 26 · 101− 35 · 75= 26 · 101− 35 · (4620− 45 · 101)
= −35 · 4620+ 1601 · 101
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 24
Extended Euclid’s Algorithm
101 · s+ 4620 · t = 1
Run Euclid’s algorithm:
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Work backwards, to express GCD interms of a1 = 101 and a0 = 4620:
1= 3− 1 · 2= 3− 1 · (23− 7 · 3) = −1 · 23+ 8 · 3= −1 · 23+ 8 · (26− 1 · 23) = 8 · 26− 9 · 23= 8 · 26− 9(75− 2 · 26) = −9 · 75+ 26 · 26= −9 · 75+ 26 · (101− 1 · 75) = 26 · 101− 35 · 75= 26 · 101− 35 · (4620− 45 · 101)
= −35 · 4620+ 1601 · 101
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 25
Extended Euclid’s Algorithm
101 · s+ 4620 · t = 1
Run Euclid’s algorithm:
a0 = 4620= 45 · 101+ 75
a1 = 101= 1 · 75+ 26
a2 = 75= 2 · 26+ 23
a3 = 26= 1 · 23+ 3
a4 = 23= 7 · 3+ 2
a5 = 3= 1 · 2+ 1
a6 = 2= 2 · 1a7 = 1
Work backwards, to express GCD interms of a1 = 101 and a0 = 4620:
1= 3− 1 · 2= 3− 1 · (23− 7 · 3) = −1 · 23+ 8 · 3= −1 · 23+ 8 · (26− 1 · 23) = 8 · 26− 9 · 23= 8 · 26− 9(75− 2 · 26) = −9 · 75+ 26 · 26= −9 · 75+ 26 · (101− 1 · 75) = 26 · 101− 35 · 75= 26 · 101− 35 · (4620− 45 · 101)
= −35 · 4620+ 1601 · 101
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 26
Extended Euclid’s Algorithm
−35 · 4620+ 1601 · 101= 1
Bezout coefficients are s = 1601 and t = −35.
Therefore, s = 1601 is the multiplicative inverse:
101 · 1601≡ 1 (mod 4620)
It works, but it’s easy to make a mistake using this method. Let’sdescribe the extended Euclid’s algorithm more systematically.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 27
Extended Euclid’s Algorithm (II)
The task:
Given two numbers a0 ≥ a1, run Euclid’s agorithm, computing
a2 = . . .
a3 = . . .
. . .
ak = gcd(a0, a1)
In addition, find the coefficients xk and yk such that
ak = xka0 + yka1
We find a recurrent solution for xk and yk.
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 28
Extended Euclid’s Algorithm (II)Need to find the coefficients xk and yk such that
ak = gcd(a0, a1) = xka0 + yka1
But we compute more than that. We want to represent all ai as alinear combination of a0 and a1
a0 = x0a0 + y0a1
a1 = x1a0 + y1a1
a2 = x2a0 + y2a1
a3 = x3a0 + y3a1
. . .
ak = gcd(a0, a1) = xka0 + yka1
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DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 29
Extended Euclid’s Algorithm (II)
a0 = x0a0 + y0a1
a1 = x1a0 + y1a1
a2 = x2a0 + y2a1
a3 = x3a0 + y3a1
. . .
ak = xka0 + yka1
The other x i and yi can be derived using the relations between ai ’s:
ai = ai−2 − qi−1 · ai−1
![Page 30: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/30.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 30
Extended Euclid’s Algorithm (II)
a0 = x0a0 + y0a1
a1 = x1a0 + y1a1
a2 = x2a0 + y2a1
a3 = x3a0 + y3a1
. . .
ak = xka0 + yka1
a0 = 1a0 + 0a1, x0 = 1, y0 = 0,
a1 = 0a0 + 1a1, x1 = 0, y1 = 1,
The other x i and yi can be derived using the relations between ai ’s:
ai = ai−2 − qi−1 · ai−1
![Page 31: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/31.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 31
Extended Euclid’s Algorithm (II)
a0 = x0a0 + y0a1
a1 = x1a0 + y1a1
a2 = x2a0 + y2a1
a3 = x3a0 + y3a1
. . .
ak = xka0 + yka1
a0 = 1a0 + 0a1, x0 = 1, y0 = 0,
a1 = 0a0 + 1a1, x1 = 0, y1 = 1,
The other x i and yi can be derived using the relations between ai ’s:
ai = ai−2 − qi−1 · ai−1
![Page 32: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/32.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 32
Extended Euclid’s Algorithm (II)Euclid’s algorithm computes the next remainder, ai, this way:
ai = ai−2 − qi−1 · ai−1
Two previous remainders are
ai−2 = x i−2a0 + yi−2a1 and ai−1 = x i−1a0 + yi−1a1
ai = ai−2 − qi−1 · ai−1
= x i−2 · a0 + yi−2 · a1 − qi−1(x i−1 · a0 + yi−1 · a1)= (x i−2 − qi−1 x i−1) · a0 + (yi−2 − qi−1 yi−1) · a1
=�
x i−2 −�ai−2 − ai
ai−1
�
x i−1
︸ ︷︷ ︸
=x i
�
· a0 +�
yi−2 −�ai−2 − ai
ai−1
�
yi−1
︸ ︷︷ ︸
=yi
�
· a1
![Page 33: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/33.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 33
Extended Euclid’s Algorithm (II)ai = ai−2 − qi−1 · ai−1
This is how we compute all x i and yi up to xk and yk:
x0 = 1 y0 = 0x1 = 0 y1 = 1. . . . . .
x i = x i−2 −�ai−2 − ai
ai−1
�
︸ ︷︷ ︸
=qi−1
x i−1 yi = yi−2 −�ai−2 − ai
ai−1
�
︸ ︷︷ ︸
=qi−1
yi−1
. . . . . .
In the end, we get two numbers xk and yk, so we can express theGCD as a linear combination of a0 and a1:
gcd(a0, a1) = ak = xk · a0 + yk · a1
![Page 34: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/34.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 34
Extended Euclid’s Algorithm (II)x i = x i−2 −�ai−2 − ai
ai−1
�
︸ ︷︷ ︸
=qi−1
x i−1 yi = yi−2 −�ai−2 − ai
ai−1
�
︸ ︷︷ ︸
=qi−1
yi−1
i ai q x i yi
0 a0 = 4620 - 1 01 a1 = 101 - 0 1
2 4620= 45 · 101+ 75a2 = 75 45 1− 45 · 0= 0− 45 · 1=
1 −45
![Page 35: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/35.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 35
Extended Euclid’s Algorithm (II)x i = x i−2 −�ai−2 − ai
ai−1
�
︸ ︷︷ ︸
=qi−1
x i−1 yi = yi−2 −�ai−2 − ai
ai−1
�
︸ ︷︷ ︸
=qi−1
yi−1
i ai q x i yi
0 a0 = 4620 - 1 01 a1 = 101 - 0 1
2 4620= 45 · 101+ 75a2 = 75 45 1− 45 · 0= 0− 45 · 1=
1 −45
3 101= 1 · 75+ 26 0− 1 · 1= 1− 1 · (−45) =a3 = 26 1 −1 46
4 75= 2 · 26+ 23 1− 2 · (−1) = −45− 2 · 46=a4 = 23 2 3 −137
![Page 36: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/36.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 36
Extended Euclid’s Algorithm (II)i ai q x i yi
0 a0 = 4620 - 1 01 a1 = 101 - 0 12 a2 = 75 45 1 −453 a3 = 26 1 −1 464 a4 = 23 2 3 −137
5 26= 1 · 23+ 3 −1− 1 · 3= 46− 1 · (−137) =
a5 = 3 1 −4 183
6 23= 7 · 3+ 2 3− 7 · (−4) = −137− 7 · 183=
a6 = 2 7 31 −1418
7 3= 1 · 2+ 1 −4− 1 · 31= 183− 1 · 1418=
a7 = 1 1 −35 1601
![Page 37: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/37.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 37
Extended Euclid’s Algorithm (II)
While computing the sequence of ai ’s with Euclid’s algorithm, weeventually produced coefficients
x7 = −35, y7 = 1601
By construction, they satisfy the equation
a7 = x7 · a0 + y7 · a1
1= −35︸︷︷︸
=x7
·4620︸︷︷︸
=a0
+1601︸︷︷︸
=y7
· 101︸︷︷︸
=a1
But from the last equation we can find the inverse of 101 modulo4620, and the inverse of 4620 modulo 101.
![Page 38: ModularArithmetic - GitHub PagesModulararithmetic Congruence Modulararithmetic Multiplicativeinverse ExtendedEuclid’s Algorithm p.2 Previously,wedefined Def(Divisibility).We say](https://reader034.fdocuments.us/reader034/viewer/2022042023/5e7b99d60767986b125fb426/html5/thumbnails/38.jpg)
DefinitionsFundamentaltheorem of arithmeticGCD is a linearcombinationRelative primesModular arithmeticCongruenceModular arithmeticMultiplicative inverseExtended Euclid’sAlgorithm
p. 38
Finding amultiplicative inverse
Take this equation and find the multuiplicative inverse of a1 = 101modulo a0 = 4620.
1= −35︸︷︷︸
=x7
·4620︸︷︷︸
=a0
+1601︸︷︷︸
=y7
· 101︸︷︷︸
=a1
1601 · 101− 1= 35 · 4620
Therefore, by definition of congruence,
101 · 1601≡ 1 (mod 4620).
So, 1601 is a multiplicative inverse of 101 modulo 4620.
We were able to find the inverse, because 101 and 4620 are relativeprimes, that is, their GCD is equal to 1.