1.
Arif Ahmed
NIT Silchar
12-25-109
Number Theory
arifch2009@gmail.com
2.
CONTENTS…
Why Number Theory?
The Euclidean Algorithm
Modular Arithmetic & Properties
Additive/Multiplicative Inverse
Group, Ring & Fields
Overview of GF(pn)
arifch2009@gmail.com
3.
The Euclidean Algorithm
An efficient way to find the GCD(a,b)
uses theorem that:
GCD(a,b) = GCD(b, a mod b)
Euclidean Algorithm to compute GCD(a,b) is:
EUCLID(a,b)
1. A = a; B = b
2. if B = 0 return A = gcd(a, b)
3. R = A mod B
4. A = B
5. B = R
6. goto 2
Note : If GCD(a,b)=1 , the a and b are Relatively Prime Number
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4.
Example
gcd(68, 26)
68 = 2 x 26 + 16 gcd(26,(68mod26)16)
26 = 1 x 16 + 10 gcd(16, 10)
16 = 1 x 10 + 6 gcd(10, 6)
10 = 1 x 6 + 4 gcd(6, 4)
6 = 1 x 4 + 2 gcd(4, 2)
4 = 2 x 2 + 0 gcd(2, 0)
GCD(26,16)=2
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5.
Modular Arithmetic
Define modulo operator “a mod n” to be remainder
when a is divided by n
a=qn+r 0<=r<n r=a mod n
a=11 n =7 r=4
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6.
Congruent Module n
Two Integers a, b are congruent module n
a b mod n
23 11 mod 4
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7.
Operation On modular Arithmetic
Addition , Subtraction & Multiplication Operation
can be done. They exhibits the following
properties :
1. [(a mod n) + (b mod n)] mod n = (a+b) mod n
2. [(a mod n) - (b mod n)] mod n = (a-b) mod n
3. [(a mod n) * (b mod n)] mod n = (a*b) mod n
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10.
Additive/Multiplicative Inverse
Additive Inverse of x is y if (x+y) mod n =0
So additive inverse of 1 is 7.
Multiplicative Inverse of x is y if (x*y) mod n =1
So Multiplicative e inverse of 1 is 1.
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11.
Properties of Modular Arithmetic
Zn = set of integer less then n Zn={0,1,….,(n-1)}
This class is cal Residue Class/Set of Residue. For
Z3= {0,1,2}
The Residue Class (mod 3) are
[0] ={……,-12,-9,-6,-3,0,3,6,9,……….}
[1] ={……,-11,-8,-5,-2,1,4,7,10,……….}
[2] ={……,-10,-7,-4,-1,2,5,8,11,……….}
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12.
Properties of Modular Arithmetic
Properties of Modular Arithmetic On Zn
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13.
Extended Euclidean Algorithm
Simple Euclidean: GCD(a,b) = GCD(b, a mod b)
=d
But in Extended Euclidean Algorithm, not only
calculate d, but also two additional integers x and
y such that, a*x+b*y=d
NOTE : Very Important in the arear of finite field in RSA
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15.
Group, Ring & Fields
A group G, sometimes denoted by {G, ·} is a set of
elements with a binary operation, denoted by ·,
(A1) Closure: If a and b belong to G, then a · b is also
in G.
(A2) Associative: a · (b · c) = (a · b) · c for all a, b, c
in G.
(A3) Identity element: There is an element e in G
such that a · e = e · a = a for all a in G.
(A4) Inverse element: For each a in G there is an
element a' in G such that a · a' = a' ·a = e.
arifch2009@gmail.com
16.
Abelian Group
A GROUP is abelian if it has this propwerty
(A5) Commutative: a · b = b · a for all a, b in
G.
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17.
Rings
A ring R, sometimes denoted by {R, +, x}, is a set of
elements with two binary operations, called addition
and multiplication.
Properties:
R is Abelian Group with Addition(A1-A15)
(M1) Closure under multiplication: If a and b belong
to R, then a, b is also in R.
(M2) Associativity of multiplication: a(bc) = (ab)c
for all a, b, c in R.
(M3) Distributive laws: a(b + c) = ab + ac for all a, b,
c in R.
(a + b)c = ac + bc for all a, b,
c in R.arifch2009@gmail.com
18.
Commutativity of Multiplication
A Ring is to be commutative if it satisfies the
following condition,
(M4) Commutativity of multiplication: ab = ba
for all a, b in R.
arifch2009@gmail.com
19.
Integral Domain
Integral Domain is a commutative Ring that obeys
the following Axioms
(M5) Multiplicative identity: There is an element 1 in
R such that a1 = 1a = a for all a in R.
(M6) No zero divisors: If a, b in R and ab = 0, then
either a = 0 or b = 0.
arifch2009@gmail.com
20.
Field
A Field F, sometimes denoted by {F, +, x}, is a set of
elements with two binary operations, called addition and
multiplication
1. F is an integral domain
(M7) Multiplicative inverse: For each a in F, except
0, there is an element
a-1 in F Such that a*(a-1)=(a-1)*a=1
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22.
Finite Field of the Form GF(p^n)
Infinite fields are not of particular interest in the
context of cryptography.
However, finite fields play a crucial role in many
cryptographic algorithms.
It can be shown that the order of a finite field (number
of elements in the field) must be a positive power of a
prime, & these are known as Galois fields & denoted
GF(p^n).
in particular often use the fields:
GF(p)
GF(2n)
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23.
Galois Fields GF(p)
GF(p) is the set of integers {0,1, … , p-1} with
arithmetic operations modulo prime p
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