Number Theory and Its Applications in Cryptography

Number Theory & its Applications in
Cryptography
By
Kapil N Hande
Assistant Professor in CSE
Assistant Professor in CSE
Priyadarshini Bhagwati College of Engineering, Nagpur.

Contents
• Introduction
• Number Theory Basics
• Modular Arithmetic
• Algorithms & Theorems
2/3/2025 Kapil Hande 2
• Algorithms & Theorems
• Conclusion

Introduction
• Cryptography -- from the Greek for “secret
writing” -- is the mathematical “scrambling” of
data so that only someone with the necessary
key can “unscramble” it.
• Cryptography allows secure transmission of
private information over insecure channels (for
example packet-switched networks).
• Cryptography also allows secure storage of
sensitive data on any computer.

Basic Terms
• Cryptology (to be very precise)
– Cryptography --- code designing
– Cryptanalysis --- code breaking
• Cryptologist:
– Cryptographer & cryptanalyst
– Cryptographer & cryptanalyst
• Encryption/encipherment
– Scrambling data into unintelligible to
unauthorised parties
• Decryption/decipherment
– Un-scrambling

Types of Ciphers
• Private key cryptosystems/ciphers
– The secret key is shared between two parties
• Public key cryptosystems/ciphers
– The secret key is not shared and two parties
– The secret key is not shared and two parties
can still communicate using their public keys

Overview

Mathematics of Cryptography
• Integer Arithmetic
• Set of integers, denoted by Z, contains all integral
numbers, negative infinity to positive infinity
...}
,.........
2
,
1
,
0
,
1
,
2
..,
{......... 


Z
• Three binary operations addition, subtraction
and multiplication
• Division is relation but it produces two output
a = q x n + r

Integer Arithmetic
Fact 1: The integer 1 has only one division,
itself
Fact 2: Any positive integer has atleast two
Fact 2: Any positive integer has atleast two
divisors, 1 and itself.

Modular Arithmetic
• In a = q x n + r we are interested in only
one of the outputs, the remainder r
• This implies that we can change the above
relation into a binary operator with two
relation into a binary operator with two
inputs a and n and one output r
• This operator is called modulo operator
• The second input n is called modulus
• The output r is called residue

Example
• a mod n = r
• 27 mod 5 = 2
• 35 mod 12 = 0
• -18 mod 14 = 10
• -18 mod 14 = 10
• -7 mod 10 = 3

Modular Arithmetic
• “remainders-only” arithmetic
• Addition, subtraction, multiplication,
division relative to a modulus n
• Numbers typically between 0 and n-1
Notation:
• x mod n means “remainder after dividing
x by n”
• x  y (mod n) means “x and y have the
same remainder mod n”

Congruence
• Concept of congruence instead of equality
• Infinite members from Z can map to one
members of Zn
• E.g. 2 mod 10=2, 12 mod 10=2,
• E.g. 2 mod 10=2, 12 mod 10=2,
22 mod 2=2
• So 2, 12, 22 are called congruent mod 10
• We need to use congruence operator, Ξ

Congruence
• We add the phrase (mod n) to the right
side of the congruence to define the value
of modulus that makes the relationship
valid.
Important:
Important:
a and b are said to be congruent modulo n,
if their difference a-b is an integer multiple
of n
i.e. Both numbers have same remainder
when divided by n

Example
• 2 Ξ 12(mod 10)
• 13 Ξ 23(mod 10)
• 34 Ξ 24(mod 10)
• - 8 Ξ 12(mod 10)
• - 8 Ξ 12(mod 10)
• 3 Ξ 8 (mod 5)
• 8 Ξ 13(mod 5)
• 23 Ξ 33(mod 5)
• - 8 Ξ 2 (mod 5)

Congruence
• Equality operator maps a member Z to
itself, whereas congruence operator maps
a member from Z to a member Zn
• Equality operator is one-to-one
• Equality operator is one-to-one
• Congruence operator is many-to-one
• Phrase (mod n) is indication of destination
set Zn

Modular Addition
• 7 + 9 = 16
• (7 + 9) mod 10 = 6
• 7 + 4 = 11
• 7 + 4 = 11
• (7 + 4) mod 10 = 1
• 5 + 5 = 10
• (5 + 5) mod 10 = 0

Modular Subtraction
• 7 - 9 = -2
• (7 - 9) mod 10 = 8
• 7 - 4 = 3
• 7 - 4 = 3
• (7 - 4) mod 10 = 3
• 5 - 5 = 0
• (5 - 5) mod 10 = 0

Modular Multiplication
• 7 × 9 = 63
• (7 × 9) mod 10 = 3
• 7 × 4 = 28
• 7 × 4 = 28
• (7 × 4) mod 10 = 8
• 5 × 5 = 25
• (5 × 5) mod 10 = 5

Modular Division
• 7 ÷ 9 = 7/9
• (7 ÷ 9) mod 10 = 3
proof: 3 × 9 mod 10 = 7
• 7 ÷ 4 = 7/4
• (7 ÷ 4) mod 10 = ??
• (7 ÷ 4) mod 10 = ??
undefined: no x such that 4 × x mod 10 = 7
• 5 ÷ 5 = 1
• (5 ÷ 5) mod 10 = ??
• undefined: more than one x such that 5 ×
x mod 10 = 5

Greatest Common Divisor GCD
• Only one GCD possible for two numbers
• The greatest common divisor (GCD) of
two numbers is the largest number that
evenly divides both
– Two numbers are relatively prime if GCD = 1
– Two numbers are relatively prime if GCD = 1
• Examples:
– GCF (9, 10) = 1
– GCF (4, 10) = 2
– GCF (5, 10) = 5

Greatest Common Divisor GCD
• often want no common factors (except 1)
and hence numbers are relatively prime
– eg GCD(8,15) = 1
– hence 8 & 15 are relatively prime
– hence 8 & 15 are relatively prime

Euclid’s 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

Example GCD(1970,1066)
1970 = 1 x 1066 + 904 gcd(1066, 904)
1066 = 1 x 904 + 162 gcd(904, 162)
904 = 5 x 162 + 94 gcd(162, 94)
162 = 1 x 94 + 68 gcd(94, 68)
94 = 1 x 68 + 26 gcd(68, 26)
94 = 1 x 68 + 26 gcd(68, 26)
68 = 2 x 26 + 16 gcd(26, 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)

Prime Numbers
• Asymmetric key cryptography uses primes
extensively
• Positive integers have three groups, the
number 1, primes and composites
number 1, primes and composites
• Smallest prime is number 2
• Two positive integers, a and b, are
relatively prime, or co-prime, if gcd(a,b)=1

Prime Numbers
• If P is prime, then all integers 1 to P-1 are
relatively prime to P
• Numbers of prime is infinite
• π(n) finds the number of primes smaller
• π(n) finds the number of primes smaller
than or equal to n
• π(1)=0, π(2)=1, π(3)=2, π(10)=4, π(20)=8,
π(100)=25
• If n is very large, we can only use
approximation

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
(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)

Euler Totient Function ø(n)
• to compute ø(n) need to count number of
residues to be excluded
• in general need prime factorization, but
– for p (p prime) ø(p) = p-1
– for p (p prime) ø(p) = p-1
– for p.q (p,q prime) ø(pq) =(p-1)x(q-1)
• eg.
ø(37) = 36
ø(21) = (3–1)x(7–1) = 2x6 = 12

Euler's Theorem
• a generalisation of Fermat's Theorem
• aø(n) = 1 (mod n)
– for any a,n where gcd(a,n)=1
• eg.
• 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

RSA ALGORITHM
• Block Cipher, plaintext and ciphertext are
integers between 0 and n-1 for some n
• n is 1024 bits or 309 decimal digits
• Uses an expression with exponentials
• Uses an expression with exponentials
• M – Plaintext block C – Ciphertext block
• C = Me mod n
• M = Cd mod n = (Me)d mod n =Med mod n

RSA ALGORITHM
• Sender knows value of e, and only
receiver knows value of d
• So public key is PU={e,n}
• Private key is PR={d,n}
• Private key is PR={d,n}
• It is relatively easy to calculate Me mod n
and Cd mod n for all M<n
• It is infeasible to determine d given e and
n

RSA ALGORITHM
• p,q two primes (private & chosen)
• n=pq (public & calculated)
• e, with gcd(Ø(n), e)=1,
1<e<Ø(n) (public & chosen)
1<e<Ø(n) (public & chosen)
• d Ξe-1 (mod Ø(n)) (private & calculated)

Key Generation
• Select p and q p≠q and primes
• Find n=pxq
• Find Ø(n)=(p-1)(q-1)
• Select e such that gcd(Ø(n),e)=1
• Select e such that gcd(Ø(n),e)=1
1<e<Ø(n)
• Find d d Ξe-1 (mod Ø(n))
• Public Key PU={e,n}
• Private Key PR={d,n}

Encryption
• M<n C = Me mod n
Decryption
Decryption
• M = Cd mod n

RSA Example
• Let p=5, q=11, e=3, M=9
• n=pq=11x5=55
• Ø(n)=(5-1)(11-1)=4x10=40
• e=3
• e=3
• So d.e Ξ1 (mod 40)
d.3 Ξ1 (mod 40)
27.3 Ξ1 (mod 40) So d=27

RSA Example
• So PU={3,55} and PR={27,55}
• C = Me mod n =93 mod 55
• C = 14
• M = Cd
mod n= 1427 mod 55
• M = Cd
mod n= 1427 mod 55
• So 1427 mod 55
=[(141 mod 55).(142 mod 55).(144 mod 55).(148 mod 55).
(148 mod 55).(144 mod 55)] mod 55
= [14x31x26x16x16x26] mod 55 = 75106304 mod 55 = 9
So M = 9 i.e. plaintext

Comparison
Algorithm Encryption/
Decryption
Digital
Signature
Key
Exchange
RSA Yes Yes Yes
Elliptic Yes Yes Yes
Elliptic
Curve
Yes Yes Yes
Diffie-
Hellman
No No Yes
DSS No Yes No

Conclusions
• Number theory has great importance in
cryptography
• Need randomness and uniqueness in
choosing large numbers
choosing large numbers
• Primes need to be used heavily
• RSA, secure and powerful algorithm for
key exchange, digital signature &
En/Decryption

References
• www.williamstallings.com
• http://ijns.femto.com.tw/
• http://www.cacr.math.uwaterloo.ca/hac/
• www.rsa.com
• www.rsa.com

Thank You!
Thank You!

Number Theory and Its Applications in Cryptography

More Related Content

Similar to Number Theory and Its Applications in Cryptography (20)

More from kapilhande1 (8)

Recently uploaded (20)

Number Theory and Its Applications in Cryptography