A Festive Ph D Lecture Eylon Yogev Search

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A Festive Ph. D Lecture Eylon Yogev

A Festive Ph. D Lecture Eylon Yogev

Search Problems: A Cryptographic Perspective

Search Problems: A Cryptographic Perspective

Moni

Moni

Computer Science: A Cryptographic Perspective

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,

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

Reverse Randomization

TFNP – TOTAL FUNCTION NP [MP 91] Megiddo. Papadimitriou FNP TFNP FP NP co.

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]

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] ?

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?

WHAT IS THE WEAKEST ASSUMPTION UNDER WHICH WE CAN SHOW HARDNESS OF TFNP?

BARRIERS FOR PROVING TFNP HARDNESS

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

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

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 Algorithmica : P = NP

THE FIVE WORLDS OF IMPAGLIAZZO TFNP Hardness Algorithmica : P = NP

THE FIVE WORLDS OF IMPAGLIAZZO TFNP Hardness Algorithmica : P = NP

PROOF {0, 1}n D Not in TFNP

PROOF {0, 1}n D Not in TFNP

REVERSE RANDOMIZATION {0, 1}n D Distributed according to D Random shift of D Not

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 D {0, 1}n

PROOF {0, 1}m U

PROOF {0, 1}m U

PROOF {0, 1}m U

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?

PROOF {0, 1}m U r L’ r Hard distribution?

IS THIS A HARD DISTRIBUTION? The instance is the random coins of the 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

FINDING A GOOD SHIFT

NISAN-WIGDERSON PSEUDORANDOM Assume a hard function exists GENERATOR Exists pseudorandom generator Combine all shifts

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] ?

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

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

The succinct Black-Box Model 32

MODELS OF COMPUTATION The Black. Box Model The White. Box Model Complexity: # of

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

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

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

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

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

Collision resistant hash / one-way permutation NP is hard-onaverage ? Ramsey Obfuscation/FE 38