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This document describes a simple RSA encryption and decryption system implemented in code, detailing functions for modular exponentiation, modular inverses, GCD calculation, and RSA key generation. It outlines the code structure and key functions, including how keys are generated and how messages are encrypted and decrypted. The document also includes a sample output demonstrating the functionality of the RSA system with specific prime numbers and a message.

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7 views2 pages

Report Project 1-2

This document describes a simple RSA encryption and decryption system implemented in code, detailing functions for modular exponentiation, modular inverses, GCD calculation, and RSA key generation. It outlines the code structure and key functions, including how keys are generated and how messages are encrypted and decrypted. The document also includes a sample output demonstrating the functionality of the RSA system with specific prime numbers and a message.

Uploaded by

comedykhomedy
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as TXT, PDF, TXT or read online on Scribd
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This code implements a simple RSA encryption and decryption system,which includes

functions for modular


exponentiation, computing modular inverses,calculating the GCD, and generating RSA
keys.

1. Code Structure :

mod_expo: This function performs modular exponentiation, which calculates a^b(mod


n) efficiently
This is crucial for both encryption and decryption, as large powers of numbers need
to be reduced modulo
n in RSA operations.

mod_inv: Uses the Extended Euclidean Algorithm to find the modular inverse of a
under modulo phi.This is
used to calculate the private key d in RSA, where d is the modular inverse of e
modulo phi.

gcd: Calculates the GCD (Greatest Common Divisor) of two numbers. This is used to
ensure that e(public exponent)
is coprime to phi in the RSA setup.

generatekeys: Generates the RSA public and private keys. It calculates n=p×q and
ϕ=(p−1)×(q−1),
then finds an appropriate e such that 1<e<ϕ and gcd(e,ϕ)=1. Finally, it calculates
d as the modular inverse of
e modulo phi.

main: Drives the program by setting prime numbers p and q, generating keys, and
performing encryption and
decryption on a message.

2. Explanation of Key Functions :

mod_expo(ll a, ll b, ll mod) :This function uses the "Exponentiation by Squaring"


method to compute a^b(mod n)
ans accumulates the result.
The loop iteratively squares a and reduces b until b=0, reducing each intermediate
result by mod to prevent
overflow.

mod_inv(ll a, ll phi) :Uses the Extended Euclidean Algorithm to find the modular
inverse of a under modulo phi.
The modular inverse is necessary to find d in RSA, ensuring that d×e≡1modϕ.

gcd(ll a, ll b) :Implements the Euclidean Algorithm for finding the GCD, which
ensures that e and phi are
coprime. It is crucial for ensuring that the selected public exponent e is valid.

generatekeys(ll p, ll q, ll *n, ll *e, ll *d) : This function generates RSA keys:-


-Calculates n and phi.
-Iteratively finds an appropriate value for e by ensuring gcd(e,ϕ)=1.
-Computes the private exponent d as the modular inverse of e modulo phi.

main :
In main, the program initializes primes p and q, generates keys, and then encrypts
and decrypts a sample
message (ASCII 65 for 'A'):
Encrypts the message using message^e(mod n)
Decrypts using encrypted_message^d(mod n)

3.Sample Output :

with p=61,q=53,message='A'(ASCII value of A) :-


Public Key (n, e): (3233, 7)
Private Key (n, d): (3233, 1783)
Original Message: 65
Encrypted Message: 2790
Decrypted Message: 65

4.Summary :

This code effectively demonstrates RSA encryption and decryption in a simplified


form. Key areas for
improvement include adding error handling, larger default values for e, and
enhanced security practices
for practical use.

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