Arif Ahmed
NIT Silchar
12-25-109
Number Theory
arifch2009@gmail.com
CONTENTS…
 Why Number Theory?
 The Euclidean Algorithm
 Modular Arithmetic & Properties
 Additive/Multiplicative Inver...
The Euclidean Algorithm
 An efficient way to find the GCD(a,b)
 uses theorem that:
 GCD(a,b) = GCD(b, a mod b)
 Euclid...
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 ...
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=1...
Congruent Module n
 Two Integers a, b are congruent module n
a b mod n
23 11 mod 4
arifch2009@gmail.com
Operation On modular Arithmetic
Addition , Subtraction & Multiplication Operation
can be done. They exhibits the following...
Additive/Multiplicative Inverse
+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 3 4 5 6 7 0 1 2
...
Multiplication Module of 8
* 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 6 0 2 4 6
3 0 3 6 1 4 7 2 5
4 0 4...
Additive/Multiplicative Inverse
 Additive Inverse of x is y if (x+y) mod n =0
So additive inverse of 1 is 7.
 Multiplica...
Properties of Modular Arithmetic
 Zn = set of integer less then n Zn={0,1,….,(n-1)}
This class is cal Residue Class/Set o...
Properties of Modular Arithmetic
Properties of Modular Arithmetic On Zn
arifch2009@gmail.com
Extended Euclidean Algorithm
 Simple Euclidean: GCD(a,b) = GCD(b, a mod b)
=d
But in Extended Euclidean Algorithm, not on...
Example
arifch2009@gmail.com
Group, Ring & Fields
 A group G, sometimes denoted by {G, ·} is a set of
elements with a binary operation, denoted by ·,
...
Abelian Group
 A GROUP is abelian if it has this propwerty
(A5) Commutative: a · b = b · a for all a, b in
G.
arifch2009@...
Rings
 A ring R, sometimes denoted by {R, +, x}, is a set of
elements with two binary operations, called addition
and mul...
Commutativity of Multiplication
 A Ring is to be commutative if it satisfies the
following condition,
 (M4) Commutativit...
Integral Domain
 Integral Domain is a commutative Ring that obeys
the following Axioms
 (M5) Multiplicative identity: Th...
Field
 A Field F, sometimes denoted by {F, +, x}, is a set of
elements with two binary operations, called addition and
mu...
arifch2009@gmail.com
Finite Field of the Form GF(p^n)
 Infinite fields are not of particular interest in the
context of cryptography.
 Howeve...
Galois Fields GF(p)
 GF(p) is the set of integers {0,1, … , p-1} with
arithmetic operations modulo prime p
arifch2009@gma...
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Introduction to Number theory

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Introduction to Number theory

  1. 1. Arif Ahmed NIT Silchar 12-25-109 Number Theory arifch2009@gmail.com
  2. 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. 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 arifch2009@gmail.com
  4. 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 arifch2009@gmail.com
  5. 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 arifch2009@gmail.com
  6. 6. Congruent Module n  Two Integers a, b are congruent module n a b mod n 23 11 mod 4 arifch2009@gmail.com
  7. 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 arifch2009@gmail.com
  8. 8. Additive/Multiplicative Inverse + 0 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 0 2 2 3 4 5 6 7 0 1 3 3 4 5 6 7 0 1 2 4 4 5 6 7 0 1 2 3 5 5 6 7 0 1 2 3 4 6 6 7 0 1 2 3 4 5 7 7 0 1 2 3 4 5 6 Arithmetic Module of 8 arifch2009@gmail.com
  9. 9. Multiplication Module of 8 * 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 0 2 4 6 0 2 4 6 3 0 3 6 1 4 7 2 5 4 0 4 0 4 0 4 0 4 5 0 5 2 7 4 1 6 3 6 0 6 4 2 0 6 4 2 7 0 7 6 5 4 3 2 1 Multiplication Module of 8 arifch2009@gmail.com
  10. 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. arifch2009@gmail.com
  11. 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,……….} arifch2009@gmail.com
  12. 12. Properties of Modular Arithmetic Properties of Modular Arithmetic On Zn arifch2009@gmail.com
  13. 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 arifch2009@gmail.com
  14. 14. Example arifch2009@gmail.com
  15. 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. 16. Abelian Group  A GROUP is abelian if it has this propwerty (A5) Commutative: a · b = b · a for all a, b in G. arifch2009@gmail.com
  17. 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. 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. 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. 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 arifch2009@gmail.com
  21. 21. arifch2009@gmail.com
  22. 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) arifch2009@gmail.com
  23. 23. Galois Fields GF(p)  GF(p) is the set of integers {0,1, … , p-1} with arithmetic operations modulo prime p arifch2009@gmail.com
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