Generating Hard Np-Complete Instances If Factoring Is Np-Complete

Defining the Core Problem

We formalize the computational problem of systematically generating hard or computationally intractable instances for NP-complete decision problems under the assumption that integer factorization is an NP-complete problem. NP-complete problems have two key properties: (i) they can be verified in polynomial time, and (ii) if any NP-complete problem has a polynomial-time solution, all NP problems can be solved in polynomial time. Integer factorization requires finding the prime integer factors of a given composite integer. Constructing new NP-complete problem instances that encode integer factoring as subproblems would inherit factoring’s conjectured computational hardness.

Leveraging Factorization Hardness

The widely believed mathematical conjecture that efficiently factoring large integers is intrinsically difficult motivates an approach to create challenging NP-complete problem instances. While verified solutions to these problems could confirm a solution’s validity in polynomial time, the process of finding solutions may require submarine search techniques through an exponentially-sized search space. By embedding difficult factorization problems within larger NP-complete problems, inherent factoring difficulty manifests as hardness in solving the fully reduced problems. For example, efficiently solving Sudoku puzzles that encode 100-digit semiprime factorization may implicitly require factoring the embedded large semiprime number.

Mapping Integer Factors to NP-Complete Solutions

Methodologically, wewant systematic techniques to inject arbitrary integer factorization problems as subproblems into NP-complete problems while preserving solution verifiability. For example, we can encode a semiprime number’s factors as a satisfying Boolean assignment for a propositional formula such that recovering the assignment reveals the factors. ThroughMany-One reductions, factoring semiprime n maps to satisfiability solutions mapping one-to-one onto the factors p and q. This embedding enables translating hard factoring instances into equally hard satisfiability problems deterministically.

Generating Large Integers Resistant to Factoring

To maximize difficulty for the reduced NP-complete problem, we focus on generating semiprime numbers that are hard to factor in practice based on current algorithms and computing power. Constructing semiprimes by multiplying random large prime numbers can systematically produce integers with high bit-lengths resilient to integer factorization using the best known classical and quantum algorithms. We can quantify hardness by lower-bounding the number of bit operations required to factor based on semiprime constructions relying on well-studied computational assumptions from number theory.

Translating Factorization to Satisfiability

As a concrete example, we provide a Many-One reduction from factoring an l-bit semiprime n to a Boolean satisfiability problem such that factors p and q of n correspond to a unique satisfying assignment. First, we represent the l bit positions of n as Boolean variables. We then construct clauses encoding multiplication relationships based on each bit’s contribution to the product n, ensuring a satisfying assignment corresponds uniquely to factors p and q. Hard random semiprimes n will translate to satisfiability instances with no shortcut solving strategies beyond searching over possible factor pairs.

Benchmarking Hardness of Generated Problem Instances

To validate desired computational hardness, we need to empirically test the difficulty of solved reduced problem instances generated from factoring embeddings. Benchmarking can establish lower bounds on runtime by attempting to solve translations using the best known exhaustive and heuristic algorithms. For example, we can measure the solve time for the satisfiability formulas produced by factoring reductions to confirm increasing runtimes for larger semiprime bit-lengths. If substantially increased solve times correlate with semiprime hardness, achieved problem difficulty mirrors factoring difficulty.

Optimizing Encodings for Maximum Hardness

While a naive reduction may encode factors into an NP-complete problem, additional work can potentially yield equivalent problem instances that are more computationally demanding in practice. By focusing on optimizing the mapping of factoring subproblems, we can potentially increase resulting instance hardness. This can minimize structural artifacts that specialized algorithms could exploit. For example, we can minimize variable dependencies or enforce uniformity of clauses and constraints. Such refinements force reliance on general-purpose brute force search techniques without short cuts.

Case Study: Generating Tough Combinatorial Problems

As an end-to-end demonstration, we provide details for encoding difficult factoring as input for generating computationally demanding graph coloring instances. First, we generate a random 512-bit RSA modulus n to factor. We represent the factors p and q in a reduction producing a graph G with n nodes indicating variables, and edge relations indicating factors’ bit contributions to n’s product. G’s chromatic number encodes constraints on variable assignments, forcing any valid coloring to reveal factors p and q. Running benchmark graph coloring heuristics on translated outputs could validate exponential runtime scaling with n’s bit length.