Syllabus
- Motivating examples of cryptographic tasks (private key cryptography,digital signatures). Classical approach (th. of information) and modern approach (computational complexity th.).
- Bijection between words over a finite alphabet and natural numbers. Languages and decision problems. Turing machines (TM). Variants of TM (more tapes, etc.).
- Recursive function (RF) a partial recursive functions (PRF). Recursive sets (R) and recursively enumerable sets (RE). coRE sets.
- Thms: R is the intersection of RE and coRE. A set is RE iff it is the range of a PRF iff it is a projection of a recursive relation. A link between RE sets and NTMs (non-deterministic TM).
- Coding of TMs by words. Universal TM. Halting problem (HALT) and 10th Hilbert's problem. HALT is not recursive.
- Time complexity of TM and NTM. Polynomial time (p-time), class P. Class NP (the equivalence of definitions via NTM and as p-bounded projections of p-relations).
- Space complexity, classes L, NL and PSPACE. General relations between time and space complexity.
- s-t connectivity in directed graphs (the idea of an algorithm using quadratic-log space). Savitch thm: NSPACE(f) is included in SPACE(O(f^2)). Immerman - Szelepscenyi thm: NSPACE(f) = coNSPACE(f) (without prf.).
- The hierarchy of complexity classes: SPACE(f) is a proper subclass of SPACE(g), if f = o(g), TIME(f) is a proper subclass of TIME(g), if log(f) = o(g) (without prf.). Classes E and EXP. NP is included in EXP.
- Propositional logic, boolean circuits.
- p-reducibility. NP-complete languages. Cook's thm. SAT, CIRCSAT, 3SAT, 3COLOR.
- P/poly: non-uniform p-time. Characterisations using advice words and via circuits, and their equivalence.
- Probabilistic algorithms. Examples: PIT (polynomial identity testing) and random walks on graphs (s-t connectivity; without prf.).
- Classes BPP, RP, ZPP. BPP is included in EXP. Amplification of probability in RP.
- Remark on the hypothesis that P = BPP (Impagliazzo-Wigderson thm - without prf.).
- Finite probabilistic spaces, events, random variables. Expected value E(X). Linearity of E(X). Example: The number of heads in n coin tosses.
- Conditional probability, independent events and random variables. Markov's inequality (with prf.) and Chernoff's inequality (without prf.).
- Probability amplification in BPP. BPP is included in P/poly.
- Distributions and their computability. Negligible functions. Computationally indistinguishable distributions, basic properties. Poly-many independent copies.
- Pseudo-random distributions. Pseudo-random generators (PRNG) and link to the P vs. NP problem.
- Properties of PRNGs and consequences of their existence: Polynomial extendability, derandomization of BPP in sub-exponential time and pseudo-random functions (without prf.).
- One-way functions (OWF). Weak OWF and the equivalence with the definition of OWF (without prf.). A PRNG is a OWF. Examples: Prime factorization, discrete logarithm, RSA, Rabin's function.
- Thm.: The existence of a OWF implies the existence of a PRNG (without prf.). A proof of weaker statement: The existence of a OWP (permutation) implies the existence of a PRNG. Hard bit of OWF.
- Goldreich-Levin's algorithm and theorem (with prf.).
- The characterization of PRNGs using bit unpredictability (without prf.). The definition of functions hard on average.
- Interactive proof systems. Class IP and observations: NP and coRP are included in IP. Shamir thm.: IP=PSPACE (without prf.). A proof of a special case: coNP is included in IP.
- PCP proof system. PCP thm. (without prf.). Alternative formulation: Amplification of the minimal number of unsatisfied clauses in 3CNF formulas. A proof of the equivalence of the two formulations. An idea of an application: Non-approximability of optimization problems.
- Zero-knowledge proof systems. Variants: Perfect (PZK), statistical (SZK) and computational (CZK). PZK protocols for graph isomorphism and non-izomorphism. CZK protocol for all NP sets (assuming the existence of OWF): The idea of the construction of a CZK protocol for 3COLOR.
Annotation
The course gives an introduction into computational complexity, both basic topics (classes P and NP) and topics of special interest for cryptography
(probabilistic algorithms, one way functions, pseudorandom number generators, interactive proofs, zero-knowledge proofs).