Name: Zalte Sayali Pandurang PRN: 2020mtecsit002 Aim: The relevance of Euler’s Totient Function to the application of Cryptography. Euler's totient function counts the number of integers co-prime to n between 1 and n. It has various properties and a product formula. Euler's totient function is useful in cryptography as the RSA algorithm uses it to find encryption and decryption keys based on two large prime numbers.

1. Name: Zalte Sayali Pandurang
PRN: 2020mtecsit002
Aim: The relevance of Euler’s Totient Function to the application of Cryptography.

2. Euler’s Totient Function

3. Euler’s Totient Function:
Euler’s totient function, also known as phi function ϕ(n), counts the number of integers
between 1 and n inclusive , which are co-prime to n. Two numbers are co-prime if the greatest common divisor
equals 1 . 1 is considered to be co-prime to any number
Properties of Euler’s Totient Function:
• If p is a prime number, then gcd (p, q) = 1 for all 1 <= q < p. Therefore:
ϕ(p) = p – 1.
example: ϕ(5)=4
• If p is a prime number and k >= 1, then there are exactly pk / p numbers between 1 and pk that are divisible by p.
Which gives:
ϕ(p k) = pk – pk-1.
• If a and b are relatively prime, then :
ϕ(ab) = ϕ(a). ϕ(b)
example: ϕ(35)= ϕ(7). ϕ(5)

4. Mathematical representation
Euler’s Product Formula
It states:
Its equivalent representation:

5. What are Coprime numbers?
A Co-prime number is a set of numbers or integers which have only 1 as their common factor i.e.
their highest common factor will be 1.
Co-prime numbers are also known as relatively prime or mutually prime numbers.
Examples:
18 and 35 are co-prime numbers.
The factors of 18 are 1, 2, 3, 6, 9, and 18
factors of 35 are 1, 5, 7, and 35.
Since the HCF is 1, they are coprime.

6. In general we can state that Euler’s totient function is defined as the number of positive integers less than n that
are coprime to n.
Examples of Eulers’s Totient:
• ϕ(3)= 2
As numbers having GCD 1 with 3 are {1,2} and the value of ϕ is the count of elements in the set.
• ϕ(8)= 4
As numbers before 8 having GCD 1 with 8 are{1,3,5,7}
• ϕ(9)= 6
{1,2,4,5,7,8}
• ϕ(7)=6
As we have seen when n is prime ϕ(n)=n-1
• ϕ(12) which can also be further simplified as ϕ(3). Φ(22)=2.2=4
• ϕ(35)= ϕ(7). ϕ(5)=6.4=24

7. Use of Euler’s Totient function in Cryptography :
• In a Cryptosystem public keys are often encrypted using large prime numbers for security purpose .
• The RSA algorithm in cryptosystem makes use of Euler’s Totient function:
 User creates and publishes a public key based on two large prime numbers, along with an auxiliary value.
 The prime numbers are kept secret.
 Messages can be encrypted by anyone, via the public key, but can only be decoded by someone
who knows the prime numbers.

8. Euler’s Totient function in RSA
• RSA system involves choosing large prime numbers p and q ,
computing n = p.q and k = φ(n),
• Finding two numbers e and d such that ed ≡ 1 (mod k ).
The numbers n and e (the "encryption key") are released
To the public, and d (the "decryption key") is kept private.
• A message, represented by an integer m, where 0 < m < n, is encrypted by computing
S = me (mod n)
• It is decrypted by computing t = Sd (mod n).
• Euler's Theorem can be used to show that if 0 < t < n, then t = m.
The security of an RSA system would be compromised if the number n could be factored or if φ(n) could
be computed without factoring n.

9. REFERENCES :
Paper on “The RSAAlgorithm” by Evgeny Milanov (Published on 3 June 2009)
https://www.geeksforgeeks.org/eulers-totient-function/ (Accessed in March 2021)
cp-agorithms.com (Accessed in March 2021)
A generalization of the Euler’s totient function Marius Tarnauceanu (Published on December 5, 2013)