Hovedinnhold
Kurs: (Informatikk > Enhet 2
Leksjon 5: Modulær aritmetikk- Hva er modulær aritmetikk?
- Modulus operatør
- Modulus utfordring
- Kongruens modulo
- Kongruens sammenhenger
- Ekvivalente sammenhenger
- Kvotient rest teoremet
- Modulær addisjon og subtraksjon
- Modulær addisjon
- Modulus utfordring (addisjon og subtraksjon)
- Modulær multiplikasjon
- Modulær multiplikasjon
- Modulær eksponensiering
- Rask modulær eksponensiering
- Rask modulær eksponensiering
- Modulære inverse
- The Euclidean Algorithm
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Modulær addisjon og subtraksjon
Let's explore the addition property of modular arithmetic:
(A + B) mod C = (A mod C + B mod C) mod C
Eksempel:
Let A=14, B=17, C=5
Let's verify: (A + B) mod C = (A mod C + B mod C) mod C
LHS = Left Hand Side of the Equation
RHS = Right Hand Side of the Equation
LHS = Left Hand Side of the Equation
RHS = Right Hand Side of the Equation
LHS = (A + B) mod C
LHS = (14 + 17) mod 5
LHS = 31 mod 5
LHS = 1
LHS = (14 + 17) mod 5
LHS = 31 mod 5
LHS = 1
RHS = (A mod C + B mod C) mod C
RHS = (14 mod 5 + 17 mod 5) mod 5
RHS = (4 + 2) mod 5
RHS = 1
RHS = (14 mod 5 + 17 mod 5) mod 5
RHS = (4 + 2) mod 5
RHS = 1
LHS = RHS = 1
Intuition Behind Modular Addition
Observe the figure below. If we want to calculate 12+9 mod 7 we can easily go around the modular circle for a sequence of 12+9 steps clockwise (as shown in the bottom left circle).
We can take a shortcut by observing that every 7 steps we end up in the same position on the modular circle. These complete loops around the modular circle don’t contribute to our final position. We ignore these complete loops around the circle by calculating each number mod 7 (as shown in the two upper modular circles). This will give us the number of clockwise steps, relative to 0, that contributed to each of their final positions around the modular circle.
Now, we only have to go around the circle clockwise the total of the number of steps that contributed to each of numbers final position (as shown in the bottom right modular circle). This method applies, in general, to any two integers and any modular circle.
Proof for Modular Addition
We will prove that (A + B) mod C = (A mod C + B mod C) mod C
We must show that LHS=RHS
We must show that LHS=RHS
From the quotient remainder theorem we can write A and B as:
A = C * Q1 + R1 where 0 ≤ R1 < C and Q1 is some integer. A mod C = R1
B = C * Q2 + R2 where 0 ≤ R2 < C and Q2 is some integer. B mod C = R2
(A + B) = C * (Q1 + Q2) + R1+R2
B = C * Q2 + R2 where 0 ≤ R2 < C and Q2 is some integer. B mod C = R2
(A + B) = C * (Q1 + Q2) + R1+R2
LHS = (A + B) mod C
LHS = (C * (Q1 + Q2) + R1+ R2) mod C
We can eliminate the multiples of C when we take the mod C
LHS = (R1 + R2) mod C
LHS = (C * (Q1 + Q2) + R1+ R2) mod C
We can eliminate the multiples of C when we take the mod C
LHS = (R1 + R2) mod C
RHS = (A mod C + B mod C) mod C
RHS = (R1 + R2) mod C
RHS = (R1 + R2) mod C
LHS=RHS= (R1 + R2) mod C
Modular Subtraction
A very similar proof holds for modular subtraction
(A - B) mod C = (A mod C - B mod C) mod C
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