Algorithm - EncryptionRSA
Sakarya University - Computer Engineering
Cryptography Class
Step-1
Choose two prime number p and q Lets take p = 3 and q = 11
Step-2
Compute the value of n and \phi It is given as, n = p \times q and \phi = (p-1) \times (q-1)
Here in the example, n = 3 \times 11 = 33 \phi = (3-1) \times (11-1) = 2 \times 10 = 20
Step-3
Find the value of e (public key) Choose e, such that e should be co-prime. Co-prime means it should not multiply by factors of \phi and also not divide by \phi
Factors of \phi are, 20 = 5 \times 4 = 5 \times 2 \times 2 so e should not multiply by 5 and 2 and should not divide by 20.
So, primes are 3, 7, 11, 17, 19…, as 3 and 11 are taken choose e as 7
Therefore, e = 7
Step-4: Compute the value of d (private key) The condition is given as, gcd(\phi, e) = \phi x +ey = 1 where y is the value of d. To compute the value of d,
Form a table with four columns i.e., a, b, d, and k. Initialize a = 1, b = 0, d = \phi, k = – in first row. Initialize a = 0, b = 1, d = e, k = \frac{\phi}{e} in second row.
From the next row, apply following formulas to find the value of next a, b, d, and k, which is given as; a_{i} = a_{i-2} - (a_{i-1} \times k_{i-1}) b_{i} = b_{i-2} - (b_{i-1} \times k_{i-1}) d_{i} = d_{i-2} - (d_{i-1} \times k_{i-1}) k_{i} = \frac{d_{i-1}}{d_{i}}
As soon as, d = 1, stop the process and check for the below condition
if b > \phi b = b \mod \phi if b < 0 b = b + \phi
For a given example, the table will be,
A B D K 1 0 20 – 0 1 7 2 1 -2 6 1 -1 3 1 –
As in the above table d = 1, stop the process and check for the condition given for the b \therefore b = 3 To verify that b is correct, the above condition should satisfy, i.e. gcd(\phi, e) = \phi x + ey = (20 \times -1) + (7 \times 3) = 1. Hence d is correct.
Step-5: Do the encryption and decryption Encryption is given as, c = t^{e}\mod n Decryption is given as, t = c^{d}\mod n
For the given example, suppose t = 2, so;
Encryption is c = 2^{7}\mod 33 = 29 Decryption is t = 29^{3}\mod 33 = 2
Therefore in the final, p = 3, q = 11, \phi = 20, n = 33, e = 7 and d = 3