A Festive Ph D Lecture Eylon Yogev Search
- Slides: 37
A Festive Ph. D Lecture Eylon Yogev
Search Problems: A Cryptographic Perspective
Moni
Computer Science: A Cryptographic Perspective
Obfuscation, NP Secret. Sharing: [Komargodski-Moran-Naor-Pass-Rosen -Y 14], [Komargodski-Naor-Y 15], [Komargodski-Naor-Y 16] Search Problems Ramsey, Local Search, Hashing: [Hubacek-Naor-Y 17], [Hubacek-Y 17], [Komargodski-Naor-Y 18] Computation al Complexity Cryptography Distributed Algorithms Adversarial Bloom Filters: [Naor. Y 13], [Naor. Y 15] Data Structures Secure Distributed Algorithm & Derandomization: [Halevi-Ishai-Jain-Komargodski-Sahai. Y 17], [Parter-Y 18 a], [Parter-Y 18 b]
Reverse Randomization
TFNP – TOTAL FUNCTION NP [MP 91] Megiddo. Papadimitriou FNP TFNP FP NP co. NP P
TFNP LANDSCAPE Local Optimum “Every DAG has a sink” Continuous Local Optimum [DP 11] “Every function has a local optimum” Polynomial Pigeonhole Principle “Every length preserving function either has a collision or a preimage for 0” Nash Equilibrium Brouwer Fixed Points “every directed graph with an unbalanced node must have another” 9
HARDN ESS Ramsey [KNY 17] Collision resistant hash / one-way permutation [Pap 94] ? Obfuscation/FE [HY 17] Obfuscation/FE [BPR 15] 10
WHAT IS THE WEAKEST ASSUMPTION UNDER WHICH WE CAN SHOW HARDNESS OF TFNP?
BARRIERS FOR PROVING TFNP HARDNESS
THE FIVE WORLDS OF IMPAGLIAZZO Algorithmica : P = NP
THE FIVE WORLDS OF IMPAGLIAZZO Algorithmica : P = NP
THE FIVE WORLDS OF IMPAGLIAZZO Algorithmica : P = NP
OUR RESULTS TFNP* hardness can be based on any hard-on-average language in NP For example: planted clique, random SAT, etc. In particular, any one-way function. Our results show: hard-on-average TFNP problems exist in Pessiland (and beyond) * In the non-uniform setting (will elaborate later).
THE FIVE WORLDS OF IMPAGLIAZZO Algorithmica : P = NP
THE FIVE WORLDS OF IMPAGLIAZZO TFNP Hardness Algorithmica : P = NP
PROOF {0, 1}n D Not in TFNP
REVERSE RANDOMIZATION {0, 1}n D Distributed according to D Random shift of D Not a hard distribution
PROOF {0, 1}m D {0, 1}n
PROOF {0, 1}m U
PROOF {0, 1}m U
PROOF {0, 1}m U
PROOF {0, 1}m U r L’ r Hard distribution?
IS THIS A HARD DISTRIBUTION? The instance is the random coins of the distribution Can we learn anything from the random coins about the solution? Think of the planted clique vs. random SAT We need the distribution to be a “public-coin” distribution Do such distributions necessarily exist? Theorem: There exists a reduction from private-coin to public-coin [Impagliazzo-Levin Proof goes through universal one-way hash function 90]
FINDING A GOOD SHIFT
NISAN-WIGDERSON PSEUDORANDOM Assume a hard function exists GENERATOR Exists pseudorandom generator Combine all shifts to one good shift We can derandomize
HARDN ESS Ramsey [KNY 17] Collision resistant hash / one-way permutation [Pap 94] ? Obfuscation/FE [HY 17] Obfuscation/FE [BPR 15] 30
HARDN ESS Collision resistant hash / one-way permutation [Pap 94] NP is hard-onaverage Ramsey [KNY 17] ? Obfuscation/FE [HY 17] Obfuscation/FE [BPR 15] 31
The succinct Black-Box Model 32
MODELS OF COMPUTATION The Black. Box Model The White. Box Model Complexity: # of queries ? 33
MODELS OF COMPUTATION The Black. Box Model Complexity: # of queries The White. Box Model The Succinct Black-Box Model 34
THE SUCCINCT BLACK-BOX MODEL The algorithm is given oracle (black-box) query access to the function. 35
THE SUCCINCT BLACK-BOX MODEL The algorithm is given oracle (black-box) query access to the function. Explains why all black-box lower bounds require huge oracles Incorrect outside TFNP Example: point function 36
The Succinct Black-Box Model Main point: At every iteration we either find a solution or remove half of the rows. 0 1 1 0 0 0 1 1 37
Collision resistant hash / one-way permutation NP is hard-onaverage ? Ramsey Obfuscation/FE 38
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