Now, I will encrypt these message pair with the private key (e). We have shown how to encrypt and decrypt messages with RSA. We will just use public key (d) instead of private key (e). We are going to apply same procedure of encryption in decryption. We can use python pow function for encryption purposes. To encrypt a message, we will find its e-th power for modulus n. Suppose that the message we would like to encrypt is 99. We are ready to encrypt and decrypt messages when we have private and public key. So, public key will be 103 while private key was 7 for n = 143 and phi = 120. multiplicative inverse of 3 is 3 -1 and multiplication of a number with its multiplicative inverse must be 1. Notice that we set the exponent to -1 to find the multiplicative inverse. Base will be the private key, exponent will be -1 and finally modulus will be phi. It expects base, exponent and modulus arguments respectively.
We will use Python’s power function directly. However, this is much easier after Python 3.8. Normally, we can use extended Euclidean algorithm to find the multiplicative inverse. It comes with a greatest common divisor function.Īfter then, we need to find the multiplicative inverse of our private key for module phi. We can check the co-primeness of two integer with math module of python. Thereafter, we should pick a random integer that co-prime to phi. We do not want to spend that time! Private Public Key Pair Remember that n should be multiplication of two large primes. Otherwise, we can build a for loop to find positive integer numbers that co-prime to it but this requires O(n) complexity. In that way, we can find the phi with O(1) complexity. Instead of counting positive integer numbers that co-prime to it, we can find the multiplication of its multipliers’s totient function results. Herein, totient function is a multiplicative function. Even though, n is a prime number, it is the multiplication of two prime numbers. However, we picked the n as the multiplication of two prime numbers. We know that if n is a prime number, then positive integer number that co-prime to it should be (n-1). Totient function of n counts the positive integer numbers that co-prime to n. These variables are coming from the Euler’s generalization of Fermat’s Little Theorem.
Then, we will find the module n and totient n (or shortly phi) variables from p and q. If you set a non-prime number to p or q, then this program will raise an error. A prime number must not be divided by another number less than it without remainder. To have a safety net, we should check the primeness of this p and q pair. In a practical application, these should be large primes but we will use small ones in this example. The algorithm expects to pick two prime numbers first.
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