Cryptography Lecture 6/19/01 Overview
Public Key cryptography: The basic idea is to do away with the necessity of a secure key
exchange, which is necessary for all private key encryption schemes. The idea is as
follows:
1) Bob creates two keys, a public key, E and a private key D.
2) Bob posts the public key in a location that anyone can access.
The important thing here is that the knowledge of E does not compromise the value of D
in any way shape or form.
3) Now, anyone who wants to send a message to Bob encrypts it using the public key E.
4) Bob can now read the message using his private key D. However, since the value of E
gives no information as to the value to D, all others can not read the message.
The idea seems easy enough, but the difficulty is in finding some mathematical function
to use in this scheme such that the knowledge of E does not compromise the secrecy of
D. Clearly in all the other schemes we have seen, knowledge of the encrypting key all but
completely gives away the decrypting key.
RSA is such a scheme. In the next lecture, we will look how it works and its
development. For now, we need to go over some mathematical basics:
First, let’s define the Euler
φ
(phi) function:
φ
(n) = the number of integers in the set {1, 2, .
.., n-1} that are relatively prime to n.
φ
(p) = p –1 , for all prime numbers
φ
(pq) = (p-1)(q-1), where p and q are distinct primes. Here is a derivation of that result:
We want to count all values in the set {1, 2, 3, .
.., pq –1} that are relatively prime to pq.
Instead, we could count all value in the set NOT relatively prime to pq. We can list these
values:
p, 2p, 3p, .
.. , (q-1)p
q, 2q, 3q, .
.. (p-1)q
Note that each of these values are distinct. To notice this, see that no number of the first
row is divisible by q and no number on the second row is divisible by p. This ensures that
there are no repeats on both rows. since p and q are relatively prime, in order for q to be a
factor of a number on the first row, it would have to divide evenly into either 1, 2, 3, .
.. q-