The goals of this course are to:

• Understand the mathematical underpinnings of coding theory and cryptography
• Be able to encipher, decipher, cryptanalyse and use cryptography securely
• Be able to code and decode
• Understand the basics of elementary number theory
• Program a computer (at an introductory level)
• Improve your LaTeX skills
• Improve your ability to communicate mathematics and write proofs
• Improve your ability to think mathematically
• Have fun

5 Modules (around 3 weeks each):

Paradigms, History and Application (distributed through semester)

• Public and private key cryptography
• Keyspace and exhaustive search  
• Cryptographic “hard problems”  
• diffusion and confusion
• computational complexity:
  • Big Oh notation  
  • runtime analysis of algorithms  
  • polynomial, subexponential, exponential  
• Classical cryptosystems: 
  • substitution ciphers, 
  • affine cipher including cryptanalysis,
  • Hill cipher including cryptanalysis, 
  • Vigenere cipher  
  • frequency analysis and cryptanalysis of Vigenere
• History:  ancient, world wars, advent of public key cryptography & computers (e.g. enigma machine)
• Notions of security
• Basics of programming in Python/Sage    
• Digital signatures  
• Hash functions and passwords  
• Birthday attacks  
• Cryptography in current events, including NIST standardization  
• Timing attacks      
• Pseudo-random number generators 
• Implementation (e.g. how the internet uses cryptography, famous hacks)
• Miscellaneous topics (e.g. cryptocurrency, secret sharing, zero-knowledge proofs)

MODULE 1: Modular arithmetic & discrete logarithm

• modular arithmetic:
• computations
• reducing summands and factors but not exponents
• no cancellation
• equivalent conditions for invertibility with proofs
• set of invertible elements $(\mathbb{Z}/n\mathbb{Z})^*$ (definition and computation)
• euler’s phi function, definition and formula
• linear algebra mod n (invertibility of matrices)
• efficient exponentiation (successive squaring and double-and-add)
• multiplicative order (definition and computation)
• Fermat/Euler theorems (including proof)
• Using Fermat/Euler to tackle $a^{b^c}$ style problems
• primitive roots (definition)
• Computational Diffie-Hellman problem
• Discrete logarithm problem
• Diffie-Hellman key exchange
• Baby-step giant step
• Index calculus
• El Gamal encryption

MODULE 2: Euclidean algorithm, primality testing & factoring, RSA

• greatest common divisor    
• Euclidean algorithm (standard and extended)    
• Inverses modulo n    
• Chinese remainder theorem (many different statements, solving systems of equations)    
• Finding square roots using CRT    
• Euler phi function (proof of formula)    
• Primality testing (Fermat primality testing)    
• RSA cryptosystem (method, why you use an ephemeral key, basic security considerations)      
• RSA signature    
• attack on RSA
• the role of multiple square roots in factoring and Fermat factoring  
• p-1 factoring    
• Quadratic sieve    

MODULE 3: Finite fields and elliptic curve cryptography

• division algorithm and gcd, extended gcd in polynomial rings over $\mathbb{Z}/p\mathbb{Z}$  
• irreducible polynomials  
• finite fields: definition as a quotient (modding out by an irreducible polynomial)  
• listing the elements of a finite field in their simplest form and reducing more complicated expressions to these simplest forms  
• ability to do arithmetic in finite fields, including making an addition or multiplication table  
• number of elements in a finite field and which ones are units  
• possible sizes for finite fields (we did this just as a fact)  
• Finite field Fermat’s Little Theorem and the existence of a multiplicative generator (it always exists!)  
• finding the inverse of an element in a finite field  
• some general knowledge of the discrete logarithm in finite fields  
• ability to do DH key exchange and El Gamal encryption with a finite field  
• projective space (definition/elements over fields including finite fields)  
• elliptic curves and their group law, can compute
• number of elements on an elliptic curve  
• putting an elliptic curve in projective coordinates  
• elliptic curve factoring method  
• discrete logarithm (DLP) on elliptic curves 
• knowledge of attacks on DLP for elliptic curves 
• turning a message into a point on an elliptic curve 
• doing Diffie-Hellman key exchange or El Gamal encryption on an elliptic curve 
• Dual_EC_DRBG (how it works and why a relationship between points is a problem) 
• El Gamal Digital Signature on an elliptic curve (won’t test)
• post-quantum cryptography:  isogeny-based cryptography (only an overview)

MODULE 4: Quantum Aspects

• single qubit state and change of basis
• measurement (determine probabilities, collapse the state)
• Quantum Key Exchange (BB84)
• multiple qubits; how to measure multiple qubits, either one at a time or all together
• how to detect entanglement
• unitary operators (definition, ability to check if a matrix is unitary)
• applying unitary operators as quantum gates to qubit states
• big picture quantum computing:  we can compute classical functions/circuits “in parallel superposition”
• quantum fourier transform (able to write down small cases, able to compute its action on small cases)
• big picture QFT:  what it does to periodic states
• how to reduce factoring to period finding  
• Shor’s period-finding quantum algorithm  
• Basic continued fractions (how to compute for Shor)  

MODULE 5: Coding Theory & Lattice-Based Cryptography

• error correcting codes and their terminology (length, information rate, number of errors detected/corrected etc.) 
• Hamming distance and Hamming weight
• linear codes 
• decoding with syndromes (generating matrix, parity check matrix, cosets etc.) 
• Lattices and how they are represented, basis change 
• Shortest and closest vector problems 
• lattice reduction in 2d (LLL) 
• post-quantum cryptography:  Ring-LWE