Cryptography Modular Arithmetic and their application.pptx

Cryptography
Modular Arithmetic

2
• Modular arithmetic is a branch of arithmetic that deals with integers and their
remainders when divided by a fixed positive integer, which is called the modulus.
• In modular arithmetic, numbers "wrap around" when they reach the modulus value. It is
often denoted by the symbol "mod" or "%".
• Given integers , a, b, and a positive integer m (modulus), the expression a≡b(mod m)
a≡b (mod m) means that a and b have the same remainder when divided by m.
Modular Arithmetic

The modulo operation creates a set, which in modular
arithmetic is referred to as the set of least residues
modulo n, or Zn.
Set of Residues
Figure Some Zn sets
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Congruence
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● 23 mod 5 = 3
● Given integers a=17, b=29, and a positive
integer m=6 (modulus), the expression
a≡b(mod 6) means that a and b have the
same remainder when divided by 6.
● a is congruent to b modulo m, written as
a ≡ b (mod m)

6
https://courses.cs.washington.edu/courses/cse311/23sp/lectures/lecture10-mod.pdf

7
https://courses.cs.washington.edu/courses/cse311/23sp/lectures/lecture10-mod.pdf

Inverses
When we are working in modular arithmetic, we often need to
find the inverse of a number relative to an operation. We are
normally looking for an additive inverse (relative to an addition
operation) or a multiplicative inverse (relative to a multiplication
operation).
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In Zn, two numbers a and b are additive inverses of each other if
Additive
Inverse
In modular arithmetic, each integer has
an additive inverse. The sum of an
integer and its additive inverse is
congruent to 0 modulo n.
Note
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Example
Find all additive inverse pairs in Z10.
Solution
The six pairs of additive inverses are (0, 0), (1, 9), (2, 8), (3, 7), (4,
6), and (5, 5).
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In Zn, two numbers a and b are the multiplicative inverse of each
other if
Multiplicative
Inverse
In modular arithmetic, an integer may or
may not have a multiplicative inverse.
When it does, the product of the integer
and its multiplicative inverse is
congruent to 1 modulo n.
Note
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The extended Euclidean algorithm finds
the multiplicative inverses of b in Zn
when n and b are given and
gcd (n, b) = 1.
The multiplicative inverse of b is the
value of t after being mapped to Zn.
Note
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Example
Find the multiplicative inverse of 8 in Z10.
Solution
There is no multiplicative inverse because gcd (10, 8) = 2 ≠ 1. In
other words, we cannot find any number between 0 and 9 such
that when multiplied by 8, the result is congruent to 1.
Example
Find all multiplicative inverses in Z10.
Solution
There are only three pairs: (1, 1), (3, 7) and (9, 9). The numbers
0, 2, 4, 5, 6, and 8 do not have a multiplicative inverse.
13

Example
Find all multiplicative inverse pairs in Z11.
Solution
We have seven pairs: (1, 1), (2, 6), (3, 4), (5, 9), (7, 8), (9, 5), and
(10, 10).
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Different Sets
Figure Some Zn and Zn* sets
We need to use Zn when additive
inverses are needed; we need to use Zn*
when multiplicative inverses are needed.
Note
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Exercise
• Find the multiplicative inverse of 7 in Z16
• Find the multiplicative inverse of 6 in Z9
.
• Find all multiplicative inverse pairs in Z16.
16

Fermat's Theorem
• ap-1
mod p = 1
– where p is prime and gcd(a,p)=1
• also known as Fermat’s Little Theorem
• useful in public key and primality testing
-Suppose we have the prime number p=7. Now, let's
choose an integer a such that gcd(a,7)=1.
For simplicity, let's choose a=2.
• Now, applying Fermat's Little Theorem:
• 27−1
≡ 26
≡ 64 ≡ 1 (mod 7)
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Euler Totient Function ø(n)
• when doing arithmetic modulo n
• complete set of residues is: 0..n-1
• reduced set of residues is those numbers
(residues) which are relatively prime to n
– eg for n=10,
– complete set of residues is {0,1,2,3,4,5,6,7,8,9}
– reduced set of residues is {1,3,7,9}
• number of elements in reduced set of residues
is called the Euler Totient Function ø(n)
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Euler Totient Function ø(n)
• to compute ø(n) need to count number of
elements to be excluded
• in general need prime factorization, but
– for p (p prime) ø(p) = p-1
– for p.q (p,q prime) ø(p.q) = (p-1)(q-
1)
• eg.
– ø(37) = 36
– ø(21) = (3–1)×(7–1) = 2×6 = 12
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Euler's Theorem
• a generalisation of Fermat's Theorem
• aø(n)
mod n = 1
– where gcd(a,n)=1
• eg.
– a=3;n=10; ø(10)=4;
– hence 34
= 81 = 1 mod 10
– a=2;n=11; ø(11)=10;
– hence 210
= 1024 = 1 mod 11
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Modular Exponentiation
5350
mod 7
1. 350= 256+64+16+8+4+2
(350)10 =(101011110)2
2. 52 mod 7 = 25 mod 7= 4
54
mod 7 = (52
)2
mod 7 = 16 mod 7 =2
58
mod 7 =(54
)2
mod 7 = 4 mod 7 =4
516
mod 7 = (58
)2
mod 7 = 16 mod 7 =2
564
mod 7 =(516
)4
mod 7 = 16 mod 7 =2
5256
mod 7 =(564
)4
mod 7 = 16 mod 7 =2
3. 5350
mod 7
=5256+64+16+8+4+2
mod 7
= 2.2.2.4.2.4 mod 7
= 256 mod 7
= 4
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Cryptography Modular Arithmetic and their application.pptx

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