Its very important ppt for studying Number theory in cryptography

1. Number Theory
Dr. Chandresh Parekh
School of IT, AI & Cyber Security
Rashtriya Raksha University

2. Integers
• The set of integers, denoted by Z, contains all integral
numbers (with no fraction) from negative infinity to positive
infinity
The set of integers

3. Integer Division
• In integer arithmetic, if we divide a by n, we can get q and r.
• The relationship between these four integers can be shown as
a = q × n + r

4. Integer Division
• Assume that a = 255 and n = 11. We can find q = 23 and r = 2
using the division algorithm.
Finding the quotient and the remainder

5. Integer Division
Division algorithm for integers

6. Integer Division
• When we use a computer or a calculator, r and q are negative
when a is negative.
• How can we apply the restriction that r needs to be positive?
• The solution is simple, we decrement the value of q by 1 and
we add the value of n to r to make it positive

7. Integer Division
Graph of Division

8. Divisibility

9. Divisibility

10. Divisibility

11. Divisibility – Example

12. Divisibility – Euclidean Algorithm

13. Divisibility – Euclidean Algorithm

14. Divisibility – Euclidean Algorithm
Find the greatest common divisor of 25 and 60.
How to Find the Greatest Common Divisor by Using the Euclidian Algorithm – YouTube

15. Extended Euclidean Algorithm

16. Extended Euclidean Algorithm - Process

17. Extended Euclidean Algorithm
Extended Euclidean Algorithm - Example (Simplified) - YouTube
S = S1 – S2 *q
t = t1 – t2 *q

18. Extended Euclidean Algorithm

19. Extended Euclidean Algorithm

20. Extended Euclidean Algorithm - Exercise

21. Extended Euclidean Algorithm

22. Modular Arithmetic
• The division relationship (a = q × n + r) discussed in the
previous section has two inputs (a and n) and two outputs (q
and r).
• In modular arithmetic, we are interested in only one of the
outputs, the remainder r.

23. Modulo Operator
• The modulo operator is shown as mod. The second input (n)
is called the modulus. The output r is called the residue.

24. Modulo Operator
 Find the result of the following operations:
a. 27 mod 5
b. 36 mod 12
c. −18 mod 14
d. −7 mod 10

25. Modulo Operator
 Solution
a. Dividing 27 by 5 results in r = 2.
b. Dividing 36 by 12 results in r = 0.
c. Dividing −18 by 14 results in r = −4. After adding the
modulus r = 10.
d. Dividing −7 by 10 results in r = −7. After adding the
modulus to −7, r = 3.

26. Set of Residues

27. Congruence
• To show that two integers are congruent, we use the
congruence operator ( ≡ ). For example, we write:

28. Congruence

29. Congruence
Residue Classes
– A residue class [a] or [a]n is the set of integers
congruent modulo n.
– It is the set of all integers such that x=a(mod)n
– E.g. for n=5, we have five sets as shown below:

30. Congruence

31. Operation in Zn

32. Operation in Zn
• Perform the following operations (the inputs come
from Zn ):
a. Add 7 to 14 in Z15.
b. Subtract 11 from 7 in Z13.
c. Multiply 11 by 7 in Z20.

33. Operation in Zn
Solution :

34. Operation in Zn
• Perform the following operations (the inputs come
from either Z or Zn ):
a. Add 17 to 27 in Z14.
b. Subtract 43 from 12 in Z13.
c. Multiply 123 by -10 in Z19.

35. Operation in Zn
Solution :
• Add 17 to 27 in Z14.
(17+27)mod 14 = 2
• Subtract 43 from 12 in Z13.
(12-43)mod 13 = 8
• Multiply 123 by −10 in Z19.
(123 x (-10)) mod 19 = 5

36. Operation in Zn

37. Operation in Zn

38. Operation in Zn

39. Operation in Zn
• We have been told in arithmetic that the remainder of
an integer divided by 3 is the same as the remainder of
the sum of its decimal digits. In other words, the
remainder of dividing 6371 by 3 is same as dividing
17 by 3.
• Prove this claim using the properties of the mod
operator.

40. 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).

41. Additive Inverses

42. Additive Inverses
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).

43. Multiplicative Inverses

44. Multiplicative Inverses
• Find the multiplicative inverse of 8 in Z10.
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.
• Find all multiplicative inverses in Z10.
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.

45. Multiplicative Inverses
Find all multiplicative inverse pairs in Z11.
• Solution:
We have seven pairs: (1, 1), (2, 6), (3, 4), (5, 9), (7, 8),
(9, 9), and (10, 10).

46. Multiplicative Inverses
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 .

47. Multiplicative Inverses
To find Multiplicative Inverse Using Extended Euclidean Algorithm

48. Multiplicative Inverses
Multiplicative Inverse in cryptography - YouTube

49. Multiplicative Inverses

50. Multiplicative Inverses

51. Different Sets

52. Two More Sets
• Cryptography often uses two more sets: Zp and
Zp*.
• The modulus in these two sets is a prime
number.

53. Algebraic Structures
• Cryptography requires sets of integers and specific
operations that are defined for those sets.
• The combination of the set and the operations that
are applied to the elements of the set is called an
algebraic structure.
• Three common algebraic structures: groups, rings,
and fields.

54. Algebraic Structures

55. Primes

56. Checking for Primeness
Given a number n, how can we determine if n is a
prime?
– The answer is that we need to see if the number is
divisible by all primes less than √n
• Is 97 a prime?
– The floor of √97 = 9. The primes less than 9 are 2, 3,
5, and 7. We need to see if 97 is divisible by any of
these numbers. It is not, so 97 is a prime..

57. Checking for Primeness
Is 301 a prime?
– The floor of √301 = 17. We need to check 2, 3, 5, 7,
11, 13, and 17. The numbers 2, 3, and 5 do not divide
301, but 7 does. Therefore 301 is not a prime

58. Millar – Rabin Theorem
https://www.youtube.com/watch?v=gWx9aSepqBo

59. Euler’s Phi-Function
Euler’s phi-function, φ (n), which is sometimes called
the Euler’s totient function plays a very important role
in cryptography.
• The function finds the number of integers that are
both smaller than n and relatively prime to n

60. Euler’s Phi-Function

61. Euler’s Phi-Function

62. Euler’s Phi-Function

63. Euler’s Phi-Function

64. Euler’s Phi-Function

65. Euler’s Phi-Function and RSA Cryptosystem

66. General Idea

67. General Idea

68. Trapdoor One-Way Function

69. Trapdoor One-Way Function

70. Trapdoor One-Way Function

71. RSA Cryptosystem

72. Fermat’s Little Theorem

73. Fermat’s Little Theorem

74. Fermat’s Little Theorem

75. Fermat’s Little Theorem

76. Euler’s Theorem

77. Euler’s Theorem

78. Euler’s Theorem

79. Euler’s Theorem

80. Euler’s Theorem

81. Chinese Remainder Theorem

82. Chinese Remainder Theorem

83. Chinese Remainder Theorem

84. Chinese Remainder Theorem

85. Chinese Remainder Theorem

86. Chinese Remainder Theorem

87. Thank You