Factor Finding, Pcps, And Proof Complexity

Finding Factors Efficiently with Randomness

The problem of finding the prime factors of large integers is fundamental in number theory and cryptography. However, factoring large integers is known to be extremely computationally intensive. For an integer n, the best known classical factoring algorithms run in sub-exponential time in the size of n. Quantum algorithms can provide some speedup, but still require substantial resources.

This article explores how introducing randomness and probabilistic techniques allows development of far more efficient factor finding algorithms. We specifically focus on Probabilistically Checkable Proofs (PCP) and their applications in constraint satisfaction and proof complexity.

The Factor Finding Problem and Its Difficulty

The integer factoring or integer decomposition problem requires finding the prime divisors of a given composite integer n. A brute force approach checking all integers from 2 to √n for divisibility clearly requires O(√n) time. Slightly faster classical algorithms like the quadratic sieve and number field sieve provide sub-exponential runtimes of the form O(e^(c(log n)^(1/3))(log log n)^(2/3)) for some constant c.

The difficulty of factoring underlies the security of widely used RSA public-key cryptography, as recovering the factorization of the RSA modulus allows efficient decryption of ciphertexts. The integer factoring problem is also equivalent to solving the RSA problem, giving any efficient factoring algorithm the potential to break RSA.

Probabilistically Checkable Proofs (PCPs)

A Probabilistically Checkable Proof (PCP) allows verification that an arbitrary computation is correct with high probability by only inspecting a tiny fraction of the full proof. The key insight is to introduce randomness into the verification process.

Formally, a PCP proof system has a verifier which uses randomness to query only a small number of bits of an NP witness to verify the correctness of the witness. The soundness condition guarantees that if the NP statement is false, no proof will make the verifier accept with high probability. PCPs with constant query complexity allow verification using a number of queries independent of the proof size.

PCP theory forms the basis of modern hardness of approximation results. The PCP theorem states that every language in NP has a PCP verifier that uses only O(1) random queries, yet achieves arbitrarily small error probability. This surprising fact implies many optimal inapproximabilty results for constraint satisfaction problems.

PCPs for Efficient Factor Finding

The use of PCPs for integer factoring relies on reducing factoring to the problem of finding a satisfying assignment for a system of quadratic modular equations. This system has a satisfying assignment if and only if two numbers x and y exist such that xy = n. Verifying such assignments requires only a constant number of queries into the assignments due to the PCP architecture.

Avi Wigderson in 1991 first conceived of such PCP-based factoring algorithms by starting from the NP-complete problem of graph 3-coloring. Further innovations led to a fully polynomial randomized algorithm for factoring based on PCPs. While still superpolynomial, this demonstrates the ability of PCP techniques to provide exponential speedups over classical algorithms in some contexts.

Proof Complexity and PCPs

Proof complexity studies how difficult it is to prove propositional tautologies and contradictions symbolically. Areas like algebraic proof systems and resolution lower bounds have tight connections with the theory of PCP constructions.

For example, strong enough resolution lower bounds imply optimal hardness of approximation results matching PCP theorems. Lower bounds on the lengths of polynomial calculus proofs of tautologies show optimal superpolynomial separation between algebraic proof systems and bounded depth Frege systems.

Understanding these proof systems provides insight into the power of different verifiers for PCPs. The minimum proof length in a system also characterizes the query complexity possible for a PCP with perfect completeness and soundness based on that proof system. Advances in proof complexity are thus essential for optimizing PCP parameters.

Example PCP Code for Factor Finding

We present a simplified example PCP system for verifying integer factorization based on assignment checking.

// Prover generates quadratic modular equations 
// Such that f(x,y) = 0 iff xy = n  
func GenerateEquations(n int) []Equation {
  // Implement equation generation
}

// Verifier checks assignment using 3 queries
func VerifyAssignment(eqns []Equation, x int, y int) bool {
  
  // Select 3 random equations  
  q1, q2, q3 := SelectRandom(eqns, 3) 
  
  // Query assignments  
  b1 := Evaluate(q1, x, y)
  b2 := Evaluate(q2, x, y)  
  b3 := Evaluate(q3, x, y)
   
  // Accept if all evals are 0 mod n
  return (b1 + b2 + b3) % n == 0
}

This demonstrates how only 3 queries suffice to verify a satisfying assignment to the entire exponential-sized equation system with high probability. The modular constraints enforce that x and y are factors of n.

Applications of PCPs in Cryptography and Beyond

Although the exponential speedups from PCPs do not actually threaten cryptosystems like RSA yet, further improvements to PCP techniques may ultimately impact factoring algorithms. PCPs also enable formally verifying proofs of correctness of computations with very low resource costs. Machine learning models and algorithms may leverage PCP architectures to produce proofs that results satisfy certain properties.

Areas like verified computing, delegated computation, and transparent AI systems may be able to use PCPs to guarantee outputs match specified constraints. Blockchains and distributed ledgers can also integrate PCP-based verification of transaction validity. The applicability of PCPs thus spans far beyond their origins in computational complexity and cryptography.