Encryption

Introduction
A typical application of cryptography is secure communication. Informally a secure communication channel is one that provides both confidentiality and Integrity of the messages. In this section we investigate confidentiality, therefore we may assume that integrity is already guaranteed by some other means. (see section on integrity...#TODO)
We assume that two parties that want to communicate share a secret key kkk
. Prior to sending a message, the sender encrypts the message with the secret key, this produces a ciphertext that is then sent.  The receiver uses the same key to decrypt the message and recover the message.
Intuitively: A secure encryption scheme will prevent an eavesdropper to learn the content of the message since the ciphertext is unintelligible. The security requirement will be formalized later.
Formal definition
We introduce some notation first: We will use M\mathcal MM
 to denote the set of al possible message, K\mathcal KK
 is the set of all possible keys and C\mathcal CC
 is the set of ciphertexts. 
A symmetric encryption scheme E\mathcal EE
is a tuple of efficiently computable functions (KGen, Enc, Dec)(\text{KGen, Enc, Dec})(KGen, Enc, Dec)
.:
​
 Selects a key at random from the key space.
​Enc:M×K↦C\text{Enc}: \mathcal M \times \mathcal K \mapsto \mathcal CEnc:M×K↦C
. Encrypts the message mmm
 with the key kkk
 into a ciphertext c=Enc(m,k)c = \text{Enc}(m, k)c=Enc(m,k)
. Sometimes written as c=Enck(m)c = \text{Enc}_k(m)c=Enck​(m)
​
​Dec:C×K↦M×{⊥}\text{Dec}: \mathcal C \times \mathcal K \mapsto \mathcal M \times \{ \bot\}Dec:C×K↦M×{⊥}
. Decrypts the ciphertexts ccc
 with the key kkk
 into the message mmm
 or returns an error (⊥\bot⊥
). m=Dec(c,k)m = \text{Dec}(c, k)m=Dec(c,k)
. Sometimes written as m=Deck(c)m = \text{Dec}_k(c)m=Deck​(c)
​
ForE\mathcal EE
 to be useful we need that Dec(Enc(m,k),k)=m;∀m,k\text{Dec}(\text{Enc}(m,k), k) = m; \forall m,kDec(Enc(m,k),k)=m;∀m,k
. This is also called correctness. 
After all what's the point of confidentially sending a nice Christmas card to your grand children if they wont be able to read its content
TODO: security notions and examples

Elliptic Curve Cryptography

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The One Time Pad

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Outline

Introduction

Formal definition