ClassNotes - Gisselle Scotts Handout Encryption basically...

Gisselle & Scott’s Handout
Encryption basically works like this: Let M, C, K, define sets. M will be your the message space.
C is the cyphertext space. Cyphertext is what you call a message that has been encrypted. K is the
keyspace. Every e ∈ K is called a key and defines a unique bijection, denoted E
e
, also called the encryption
function, from M to C. In this manner, we know that every message m ∈ M maps to a unique cyphertext
c ∈ C and that every cyphertext will map to a unique message as well. Each d ∈ K defines a function, D
d
,
which is the counterpart to E
e
, called the decryption function. The decryption function defines a bijection
from C to M. An encryption shcheme will thus consist of a unique ecryption and decryption function, such
that D
d
= E
e
-1 , that is D
d
( E
e
) = m. In this case the key or key pair is (e,d) .
Encryption techniques are divided into two main branches: symmetric key encryption and public
key encryption. In symmetric key encryption it is computationally ‘easy’ to compute d from e and vice
versa. A classic example of a symmetric key encryption scheme is the the Shift cypher:
Suppose you want to encrypt the text FROG using the Shift cypher with e = 4. Each letter will be
shifted forward 4 places in the alphabet.
f → j
t → v
o → s
g → k
Your top seret encrypted information is now JVSK. To decrypt, simply perform the reverse
operation, (shift the letters back 4 spaces).
In public key encryption it is computationally infeasible to compute d from e (in other encryption
schemes this is not the case). The following is the scenario for a public key encryption scheme. So, say A
and B wish to communicate. A computes (e,d). A sends e to B. Anyone who wants to can see e. However,
since it is impossible to compute d knowing only e, it doesn’t matter that anyone can have access to e.
Later, B encrypts his message with e and sends it to A. A will then decrypt the message with d. RSA,
which we will be focusing on, is a particular example of a public key encryption scheme.
There are some basic concepts and theorems that are required in order to understand how RSA encryption
works:
The equivalence class of a, denoted [a] consists of all b’s | (bRa).
The equivalence class mod n of an integer [a] is the set of all b’s | b ≡ [a] mod n.
b ≡ a mod n means (b-a)/ n = Z, where Z is an integer (in other words n  (b-a). It follows that b = [a] + nZ.
If a ≡ a
1
mod n
∧
b ≡ b
1
modn. Then a + b ≡ ([a
1
+ b
1
]) mod n. Also, ab ≡ a
1
b
1
mod n.
For example, [1] mod 4 + [2] mod 4 is the same as [3] mod 4
The integers mod n, denoted z
n
, is the set of equivalence classes mod n {0,1… (n-1)}

Note that we only go up to (n-1) because once we reach n and above, the equivalence classes are equal.
Thus, [0] = [n], [1] = [n+1]… This is because:
[0] = b | b ≡ [0] mod n
[n] = c | c ≡ [n] mod n
[0] = b = 0 + nZ (where Z is an integer)
[n] = c = n + nZ
c = n(1 + Z), 1 + Z is an integer, so we can call it Z
1
Thus, c = 0 + n(Z
1
), which is the same as b.