Optical Architecture for (Restricted) Exponential Time Hard Problems

Optical Architecture for (Restricted) Exponential Time Hard Problems Nova Fandina Ben-Gurion University of the Negev, Israel Joint work with: Prof. Shlomi Dolev & Prof. Joseph Rosen Ben-Gurion University of the Negev, Israel Ben Gurion University of the Negev

Outline Searching for Hard Problem Succinct Permanent (mod p) is NEXP Time Hard Succinct Permanent (mod p) Has “Many ” Hard Instances Holographic Based Optical Architecture Ben Gurion University of the Negev

Modern Cryptographic Schemes Based on unproven complexity assumptions… what happens if P = NP ? • NEXP hard: don’t have a polynomial time solution • Hard on the average: randomly chosen instance is hard with high probability Ben Gurion University of the Negev

Succinct representation of Graphs[GW83] Small Circuit Representation output 0 1 2 3 n-1 0 1 2 n-1 log n log n Ben Gurion University of the Negev

Computational problems with succinctly represented inputs Succ_𝚷 problem: input:succinct representation of the graph G output: 𝚷 (G) [PY86] If 3SAT𝚷 then Succ_𝚷 is NEXP time hard Ben Gurion University of the Negev

Succinct Permanent Permanent problem where the summation is over all permutations σ of {1,…n} • #P - Complete [Val77] • Hard on Average as on the Worst Case [Lip91] Succinct Permanent the output can be too big | Ben Gurion University of the Negev

Succinct Permanent modulo (small) prime p input:small boolean circuit representing an integer matrix A with bounded (positive and negative) entries, prime p (in binary representation) k- constant c -constant output:perm A (mod p) NEXP hard & Big Set of Hard Instances Ben Gurion University of the Negev

Zero Succinct Permanent : input: small boolean circuit C representing integer matrix A with bounded entries output: permanent(A)==0 Zero Succinct Permanent (mod p): input: small boolean circuit C representing integer matrix A with bounded entries, small prime p output:permanent(A)==0 (mod p) Ben Gurion University of the Negev

Roadmap • Zero Succinct Permanent NEXP time hard • Zero Succinct Permanent Zero Succinct Permanent (mod p) • Zero Succinct Permanent (mod p ) ≤ Succinct Permanent (mod p) Ben Gurion University of the Negev

[PY86] • Succinct 3SAT is NEXP hard [Val77] • #3SATPermanen Zero Succinct Permanent is NEXP hard Ben Gurion University of the Negev

Roadmap • Zero Succinct Permanent NEXP time hard • Zero Succinct Permanent Zero Succinct Permanent (mod p) • Zero Succinct Permanent (mod p ) ≤ Succinct Permanent (mod p) Ben Gurion University of the Negev

C represent an integer matrix A with integer values constant |Permanent (A)| • Chinese Reminder Theorem: Permanent(A) can be computed by computing Permanent(A) modulo each prime p {p Ben Gurion University of the Negev

Define X be a set of first 2|U| primes The number of prime number in [1,x] is: The length of representation of each prime in X is polylogarithmical Ben Gurion University of the Negev

Randomized algorithm: • pick a prime p’ uniformly at random from the set X • compute Perm(A) mod p’ • if (Perm(A) mod p’ == 0) return Perm(A)==0 • else return Perm(A)!=0 If Per(A)==0 the answer is correct with probability 1 If Per(A)!=0 the answer is correct with probability > ½ Ben Gurion University of the Negev

Pick a prime p’ uniformly at random from the set X • pick p’ uniformly at random from [1, ] • while(! primality test(p’) ) p’ = pick [1, ] Primality test: AKS[04] Expected number of attempts: O(logn) Zero Succinct Permanent (mod p) is NEXP hard (via randomized reduction) Ben Gurion University of the Negev

Outline Searching for Hard Problem Succinct Permanent (mod p) is NEXP Time Hard Succinct Permanent (mod p) Has Many Hard Instances Holographic Based Optical Architecture Ben Gurion University of the Negev

Computing Permanent over Ben Gurion University of the Negev

Given an answers for (log n +1) matrices chosen at random from the set, compute an answer for matrix A in polynomial time : solve a system of equations to find Unique solution exists Computations mod p Ben Gurion University of the Negev

Optical Device for restricted Succinct Permanent( mod p ) • Solves the instances of the balanced form • Preprocessing unit: generates and records all matrices that can be represented by balanced small boolean circuits (holographic based implementation) • Optical Solver: given an instance outputs an encoded matrix. Forward matrix as an instance to the existing Permanent Solver. • Applies mod p operation to the result Ben Gurion University of the Negev

Preprocessing Procedure A A A A O O O O Ben Gurion University of the Negev

Preprocessing Procedure Ben Gurion University of the Negev

Holographic implementation Recording phase: Ben Gurion University of the Negev

Reconstruction phase: Ben Gurion University of the Negev

Conclusions Establishing a computational complexity of Succinct Permanent Problem mod p • NEXP time hard via randomized reduction • Average case complexity detect a hard instance and compose many hard instances Optical Solver device • restricted inputs • existence of the Permanent Solver Ben Gurion University of the Negev

Thank you! Ben Gurion University of the Negev

Optical Architecture for (Restricted) Exponential Time Hard Problems