## RSA encryption

### Problem 182

Published on Friday, 15th February 2008, 01:00 pm; Solved by 1595; Difficulty rating: 60%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`mod

^{e}`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 messages

`m` (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.