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ClassNotes - Gisselle Scotts Handout Encryption basically...

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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)}
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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.
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