kevinalmansa / rsa

Educational Implementation of RSA in C

RSA

Summary

Simple RSA implementation with detail to readabiltiy, meant to demonstrate how the algorithm works. The C programming language was chosen for this implementation due to the similarity in syntax to other languages.

This was written for educational purposes. This implementation is vulnerable to countless cryptanalysis attacks, and is not considered secure nor intended for use in production systems.

THIS IS NOT MEANT TO BE USED AS A FUNCTIONAL PROGRAM, BUT RATHER A GUIDE TO UNDERSTANDING RSA. IT IS ASSUMED THE CODE WILL BE READ.

Binaries

Two binaries can be built: rsa and rsa_tests. The first will produce a keypair given p and q, and a limit for the prime number generator used internally.

The RSA keypair is (e,n), (d,n).

The second binary was simply used to act as a very basic form of unit test ensuring encryption and decryption works.

Compatibility

This was written and tested on a Linux computer running Kernel 4.2. It should work on any Linux with gcc installed.

Care was taken to keep this program as simple as possible. No external libraries were used.

The Algorithm

RSA Algorithm

Based on: ((m**e)**d) % n = m e : encryption, d: decryption

Encryption: ciphertext = (message**e) % n

Decryption: (c**d == (m**e)**d == m) % n

Key Generation

p and q, two distinct prime numbers n = pq

fi is Euler's Totient Function fi(n) = fi(p) * fi(q) = (p - 1) * (q - 1) = n - (p + q - 1)

chose e, the public key so that:

  1. 1 < e < fi(n)
  2. gcd(e, fi(n)) == 1 (i.e. e and fi(n) are coprime

chose d, the private key so that:

  1. d == (e \*\* -1) % fi(n)
  ...d is the modular multiplicative inverse of e modulo(fi(n))

Compile

Create RSA binary

make

Recompile RSA binary

make re

Create test binary

make tests

Recompile tests

make re-tests

Remove object files created durring build

make clean

Remove object files and binaries

make fclean

Directory Structure

Directory / File	Description
src	Folder containing all the C source files
include	Folder containing all the C include files
Makefile	Configuration file for make command

Files (.c and .h)

primes

All functions related to the generation and verification of primes numbers.

Contains one global variable: g_prime_list used to store a list of prime numbers generated by sieve_of_eratosthenes

Function name	Description
gcd	Greatest common denominator
is_prime	Verifies if number is prime
sieve_of_eratosthenes	Generates a list of prime numbers, stored in g_prime_list

modulars

All functions related to modular arithmetic used by the RSA algorithm.

Function name	Description
modular_exponent_simple	Simple implementation of modular exponent
right_to_left	Optimized implementation of modular exponent
modular_inverse	Implementation of modular inverse

rsa

Contains the actual RSA algorithm

Function name	Description
pick_e	Simple algorithm used by rsa_keygen to select value e
rsa_keygen	Implementation of the RSA key generation algorithm
rsa_encrypt	Implementation of RSA encryption
rsa_decrypt	Implementation of RSA decryption