RSA encryption

The RSA encryption is based on the following procedure:

Generate two distinct primes p and q.Compute n=pq and φ=(p-1)(q-1). Find an integer e, 1<e<φ, such that gcd(e,φ)=1.

A message in this system is a number in the interval [0,n-1]. A text to be encrypted is then somehow converted to messages (numbers in the interval [0,n-1]). To encrypt the text, for each message, m, c=m^e mod n is calculated.

To decrypt the text, the following procedure is needed: calculate d such that ed=1 mod φ, then for each encrypted message, c, calculate m=c^d mod n.

There exist values of e and m such that m^e mod n=m.We call messages m for which m^e mod n=m unconcealed messages.

An issue when choosing e is that there should not be too many unconcealed messages. For instance, let p=19 and q=37. Then n=19*37=703 and φ=18*36=648. If we choose e=181, then, although gcd(181,648)=1 it turns out that all possible messagesm (0≤m≤n-1) are unconcealed when calculating m^e mod n. For any valid choice of e there exist some unconcealed messages. It's important that the number of unconcealed messages is at a minimum.

Choose p=1009 and q=3643. Find the sum of all values of e, 1<e<φ(1009,3643) and gcd(e,φ)=1, so that the number of unconcealed messages for this value of e is at a minimum.

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter