Requirements/prerequisites:

Duration: 21 weeks

Number of lectures per week: 3

Assessment:

End-of-year Examination: 3-hour end of year exam

Description: The course will be in two parts: Dr Purser will lecture on Cryptography for 2 hours per week, and Dr Murphy will lecture on Elliptic Curves for Cryptography for 1 hour per week.

Outline of Dr Purser's course on Cryptography

1. Introduction
   • The security of computer-based information, stored or transmitted.
   • Threats: Modification, Masquerade, Leakage, Replay, Repudiation, Traffic analysis, etc.
   • Services: Confidentiality (message, traffic), Authenticity (Integrity, Proof and Non-repudiation of Origin or Reception or Delivery, etc.)
   • Identification, Secure access management (handshakes), Biometrics, etc.
   • Secret keys versus secret algorithms.
   • Generation, storage and transmission of secret keys.
   • Examples: Symmetric encryption: Caesar’s cypher Integrity: CRC/Hash Authentication: Keyed hash, DES MAC
   • Other aspects: Steganography, Chaffing Winnowing, Threshold crypto, etc.
   • Standard attacks: Known plaintext/cyphertext; Chosen plaintext/cyphertext; Brute force.
   • Long messages, All-or-nothing transform.

2. Concepts
   • Shannon's theories: Unicity key lengths and distances, Perfect secrecy.
   • Symmetric key cryptography: Encryption and MACs (message authentication checks).
   • Asymmetric Key cryptography: Encryption and digital signatures.
   • Distribution and certification of public keys.
   • Time-stamping.
   • Trusted third parties (TTPs).
   • Anonymity

3. Symmetric/Secret Key Cryptology
   • History: Substitution, permutation, involution. Vigenere, Beaufort, Polyalphabetic, Jefferson Wheel, Wheatstone Disc, Enigma
   • DES (Data encryption standard), Triple-DES, IDEA etc.
   • The AES Project
   • Mars, Twofish, RC6, Serpent, Rijndael
   • Encryption modes: ECB, CBC, CFB, etc.
   • Integrity checks: MACs
   • Stream cyphers.
   • Statistical crypt-analysis, shift-and-correlate, etc.

4. Random numbers and sequences
   • For symmetric keys; as ideal cyphertext.
   • Random number generators: LCGs, LFBSRs and MLSs, BBS, de Bruin sequences, etc.
   • Tests for randomness: String lengths, Chi-square.

5. Asymmetric Public Key Cryptography
   • Concept and invention of public-key crypto (Ellis, Cocks) (Certification of public keys)
   • Bi-prime crypto
   • Modular arithmetic: Fermat, Euler, primitivity, totient function
   • The discrete logarithm (DL) problem
   • Diffie-Hellman and RSA
   • Rabin encryption
   • Very large integers and their implications.

6. Asymmetric system techniques
   • RSA parameters and frustrating attacks.
   • Primality testing: Rabin, Carmichael numbers
   • RSA security: order of the group.
   • Modular inverses, Euclid, continued fractions.
   • Chinese remainder theorem (CRT)
   • Speeding up the arithmetic: Karatsuba, Montgomery, small exponents.
   • Other algorithms: DSA/SHA-1 signature standard. RPK, MTI/A0, MTI/C0, MQV, Quadratic residues, Fiat-Shamir, Elgamal
   • Other techniques: Knapsack, Lucas series, elliptic curves, finite quaternions, affine maps, etc.
   • Holding private keys securely.

7. Hash functions
   • Desiderata
   • SHA-1, square-mod, MDC, RIPE-MD, RIPE-160, etc.
   • Keyed hash functions

8. More crypt-analysis
   • Differential crypt-analysis (Bihar-Shamir)
   • Linear crypt-analysis (Matsui)
   • Factorising: Fermat, the birthday paradox and Pollard Monte Carlo, Pollard (p+1).
   • Sub-exponential complexity and the use of factor bases.
   • Dixon’s method, Quadratic sieve, Continued fractions, Number field sieve.
   • The DL problem, Coppersmith et al.

Outline of Dr Murphy's sub-course on Elliptic Curves for Cryptography

This part of the course will study elliptic curves over finite fields, and their use in cryptography.

1. Overview
   • The discrete log problem for F_p
   • The discrete log problem for abelian groups
   • Elliptic curve cryptography

2. Preliminaries
   • Finite abelian groups
   • Finite fields
   • P, NP and NP-complete

3. Elliptic curves
   • The Weierstrass standard form
   • Addition
   • Isogenies

4. Elliptic curves over finite fields
   • Examples
   • Hasse's theorem
   • Shoof's algorithm