Introduction to cryptography

Basics of Cryptography

An Introduction to Theory of Cryptography

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WORK IN PROGRESS | PRIVATE USE ONLY

BASIC TERMINOLOGY AND CONCEPTS

Section I

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Terminology

• A Message (M) is a crucial piece of information • Sender (S) is the party that originates the message • Recipient (R) is the intended party of receipt for M • The medium through which M is sent, is called Transmission

Medium (T) • Usually this involves a Computer System (or System), composed of

hardware, software and data • A Vulnerability is a weakness in the security of the system • An Attack is an exploitation of a vulnerability, by an Intruder

(human/machine) who perpetrates (commonly an Outsider O)

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Properties of a Message

Confidentiality

• Only intended parties must receive M

Integrity

• Contents of M must be unchanged from S to R

Non-repudiation

• Once received M cannot be denied by R

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Main Types of Attack

1. Interception – Listening to the message while it passes from S to R – Does not stop R from getting the message M – Causes loss of confidentiality of message M

2. Interruption / Blocking – Prevents R from getting message M – Causes loss of availability of message M

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Main Types of Attack

3. Modification – Alteration of the contents of message M – R does not receive the original M sent by S – Causes loss of integrity of message M

4. Fabrication – R receives an authentic-looking message, as if it was

originated by S – Causes loss of integrity of message M

Closely related but different scenario is denial of M by S—called repudiation

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Requirements for a Successful Attack

• Method: tools, knowledge, skills • Opportunity: time and access to resources • Motivation: a reason to conduct the attack

• If any of these are denied, attack would not occur • But all three lie with the intruder, not system • Not practical to target and eliminate these

Method–Opportunity–Motivation: MOM

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Control of Attacks

• Control is the means by which an attack is stopped / prevented

• Stops a vulnerability from becoming an attack

• Control is a part of the system and is under our influence

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Cryptology, Cryptography & Cryptanalysis

• Cryptography is the science of (overt) secret writing, and its unauthorized decryption

• Cryptology = cryptography + cryptanalysis • Cryptography is the science of overt secret

writing • Cryptanalysis is the science of unauthorized

decryption of an encrypted message

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Cryptography Cryptography

Steganography

Technical Linguistic

Semagrams Open Code

Jargon Code

Cue

Concealment cipher

Null cipher

The Grille

Cryptography Proper

• Steganography is covert secret writing—only R and S know that M is being passed

• Cryptography proper is about overt secret writing—not only R and S know that an M is being passed

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MATHEMATICS OF CRYPTOGRAPHY

Section II

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Plaintext & Ciphertext

• Plaintext P is the original form of the message • Ciphertext C is the message in its encrypted form

• P and C are sequences of characters in the form

– P = <P1, P2, P3, P4, … > – C = <C1, C2, C3, C4, … >

• Usually P is written in lowercase while C is written in

uppercase

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Encryption & Decryption

• Encryption is the process of translating P into C • Decryption is the reverse process: C into P

– Encryption: C = E(P) – Decryption: P = D(C) – Satisfying, P = D(E(C))

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Character Sets

• A Vocabulary is a set of characters, V, used to formulate plaintext P, or set of characters, W, used to formulate C

• Length of a word is usually denoted in superscript – V*– set of words constructed from V – W*– set of words constructed from W – ε – the empty (null) word – Zn – the set of all words of length n, where, – Zn = {ε}Z1 Z2 … Zn | Zn Z*

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Encryption & Decryption

• An encryption X is a relation / rule / algorithm that is injective: – X ∶ V∗ ⇢ W∗ where x ↦ z ⋀ y ↦ z ⟶ (x = y)

• The converse is written X-1: – X−1: V∗ ⇠ W∗ x ↤ z iff (x ↦ y)

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Fiber, Homophones and Nulls

• Fiber of message 𝑥: (𝑥 ∈ 𝑉∗) is defined 𝐻𝑥 where – 𝐻𝑥 = 𝑦 ∈ 𝑊∗ 𝑥 ↦ 𝑦 𝑢𝑢𝑢𝑢𝑢 𝑋}

• If 𝑢(𝐻𝑥) > 1 then each 𝑦 ∈ 𝐻𝑥 is called a Homophone (same x, many y’s)

• If (∆∈ 𝐻𝑥: 𝜀 ↦ ∆ 𝑢𝑢𝑢𝑢𝑢 𝑋), that is, non-empty 𝐻𝑥 for empty word 𝜀 exist, they are called Nulls

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Homophones and Nulls help to mask character and word frequencies.

Cryptosystem

• A cryptosystem M is an N-tuple formed by

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Keys

• A Key is an external parameter that selects a subset of the encryption steps – C = E(P, KE): KE is the encryption key – P = D(C, KD): KD is the decryption key

• If KE = KD then the cryptosystem is symmetric,

otherwise asymmetric

• If KE = KD = ε then M is a keyless cipher

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Alphabets

• Number of steps in system M = |M| is known as its cardinality

• If |M|=1 then the system M is monoalphabetic, otherwise polyalphabetic

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Word Lengths and Blocks Word Length Encryption Decryption

1 Monographic Unipartite / Monopartite

2 Digraphic Bipartite

3 Trigraphic Tripartite

etc. Polygraphic Polypartite

• A Block is a word from 𝑉𝑛

that is subjected to one step from M

• If block length is 1 it is a stream cipher, otherwise it is a block cipher

• Note that in a suitable vocabulary of character n-tuples, a block encryption is simplified to a monographic encryption

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All the above assumes that each encryption step X is injunctive. If not, more than one word from V would encrypt to the same W. This, known as polyphony, is rarely seen.

The most basic encryption types are Substitution and Permutation (Transposition)

CRYPTANALYSIS Section III

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