RSA – An Example
Here is an example of the RSA scheme in action. I have chosen primes that are large
enough to be interesting but small enough that you can do all of the arithmetic with a
pen, paper and calculator.
Key Generation
1.
2.
3.
4.
5.
p = 31 q = 23 (chosen at random)
n = 31*23 = 713
r = 30*22 = 660
e = 223
(chosen at random)
d = 367
(computed using Euclids algorithm as below)
1 = 7 – 3(2)
1 = 7 – 3(9-7) = 4(7) – 3(9)
1 = 4(214-9(23))-3(9) = 4(214) – 95(9)
1 = 4(214) – 95(223 – 214) = 99(214) – 95(223)
1 = 99(660 – 2(223))-95(223)
1 = 99(660)-293(223)
so the inverse of 223 mod 660 = -293 = 367
660 = 223 * 2 + 214
223 = 214 * 1 +9
214 = 9 * 23 +7
9=7*1+2
7 = 2 * 3 +1
6. Bob’s private key = 367
Bob’s public key = (223, 713)
Encryption
Alice wishes to send Bob the message m = 439.
She computes c=439223 mod 713 = 284 (see fast exponent ion table below)
Alice sends the ciphertext 284 to Bob.
Decryption
Bob receives the ciphertext 284 from Alice.
He computes 284367 mod 713 = 439 (see fast exponentiation table below)
He knows the message is 439
439223 mod 713
y
u
1
439
439
211
652
315
36
118
683
377
98
242
98
98
335
335
284
n
223
111
55
27
13
6
3
1
0
284367 mod 713
y
u
1
284
284
87
466
439
656
211
94
315
94
118
397
377
652
242
652
98
439
n
367
183
91
45
22
11
5
2
1
0