Goals

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)

Suggested review problems (Trappe/Washington):  2nd Edition:  Section 2.13, #2, 3, 6, 13, 14.  3rd Edition:  Section 2.8, #3, 5, 11, Section 6.6 #1, 3.

Suggested review problems (Trappe/Washington):  2nd Edition:  Section 8.8, #1, 2, 3, 4.  3rd Edition:  #1,3,5.

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

Suggested review problems(Trappe/Washington):  Review the daily post exercises, and then consider 2nd Edition:  Section 3.13 #11, 13, 15, 18, 20, and Section 7.6 #1, 2, 3, 6, 7, 8, 11.  3rd Edition:  Section 3.13 #25, 27, 35, 39, 53 and Section 10.6 #1,2,3,7,9,11,16.  Warning:  the illegal epub 3rd ed is unusable as far as exercise numbering and math equations.

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  

Suggested review problems (Trappe/Washington):  Review the daily post exercises, and then consider 2nd Edition:  Section 3.13 #1, 2, 4,7, 8, 9, 10, and Section 6.8 # 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14.  3rd Edition:  Section 3.13 # 1,3,7, 13, 15, 17 and Section 9.8 # 1, 3, 5, 7, 9, 13, 15, 17, 21, 23, 25, 27. 

MODULE 3: Finite fields and elliptic curve cryptography

  • division algorithm and gcd, extended gcd in polynomial rings over $\ZZ/p\ZZ$  
  • 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)

Suggested review problems (Trappe/Washington):  Review the daily post exercises, and then consider 2nd Edition:  Section 3.13 # 33, 34; Section 16.7 #2, 5, 6 (more to come).  3rd Edition:  Section 3.13 #47, 48; Section 21.6 #3, 9,10 (additional in this edition only: 2, 4 for more practice).

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