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Laconic Zero Knowledge to Public Key Cryptography Public Key Cryptography Akshay Degwekar (MIT) Public Key Encryption (PKE) [Diffie-Hellman76, Rivest-Shamir-Adelman78, Goldwasser-Micali82] pk sk Public Key Encryption Public Key Encryption

  1. Laconic Zero Knowledge to Public Key Cryptography Public Key Cryptography Akshay Degwekar (MIT)

  2. Public Key Encryption (PKE) [Diffie-Hellman76, Rivest-Shamir-Adelman78, Goldwasser-Micali82] pk sk Public Key Encryption Public Key Encryption ct = Enc pk (m) GOAL: Construct “different” public-key encryption schemes public-key encryption schemes Lattices Number Theory

  3. What What structure+hardness implies public-key crypto? public-key crypto?

  4. Possible answers: NP-hardness No Crypto Known Some impossibility results [Brassard79, Feigenbaum-Fortnow93, Bogdanov- Some impossibility results [Brassard79, Feigenbaum-Fortnow93, Bogdanov- Trevisan03, Goldreich-Goldwasser98, AkaviaGoldreichGoldwasserMoshkovitz06] One-Way Functions Some barriers [Impagliazzo-Rudich89, Brakerski-Katz-Segev-Yerukhimovich11, Dachman-Soled16, Garg-Hajiabadi-Mahmoody-Mohammed18] SZK-hardness (SZK = Statistical Zero Knowledge) SZK-hardness (SZK = Statistical Zero Knowledge) Implies OWFs [Ostrovsky91] Many problems in SZK imply PKE

  5. Statistical Zero Knowledge (SZK) [ Goldwasser-Micali-Rackoff85 ] Completeness: Completeness: P * P V Soundness: Proof : All powerful P * Argument : Efficient P *

  6. Honest-Verifier Statistical Zero Knowledge: [ Goldwasser-Micali-Rackoff85 ] P V Simulator:

  7. Statistical Zero Knowledge NP PKE from SZK-Hardness? Seems Challenging : Discrete Log, Graph Iso have SZK proofs but no PKE known. SZK DLog DLog Need more Structure? Need more Structure? Graph Iso. Factoring LWE QR

  8. Example: Quadratic Non-Residuosity (Or: From GMR to GM) Can sample [Goldwasser-Micali82, Goldwasser-Micali-Rackoff85] hard instances (Honest-Verifier) (Honest-Verifier) w/ witnesses w/ witnesses Statistical Zero-Knowledge Proof Efficient Efficient Prover Prover talks very little

  9. Our Results: These Properties are Sufficient! ZK PROOF SYSTEM Public-Key Encryption + CRYPTO HARD LANGUAGE CRYPTO HARD LANGUAGE Implies One-Way Functions

  10. Instantiations QR DDH Our Our PKE PKE Assumption LWE Low noise LPN ABW Factoring CDH

  11. Perspective: Relaxing the Assumption ZK PROOF SYSTEM [Sahai-Vadhan03] [Sahai-Vadhan03] [HaitnerNguyenOng ReingoldVadhan03] + CRYPTO HARD LANGUAGE CRYPTO HARD LANGUAGE

  12. Characterization WEAK WEAK : soundness, completeness hold on average ZK PROOF SYSTEM Public-Key Encryption + + DISTRIBUTIONS DISTRIBUTIONS CRYPTO HARD LANGUAGE

  13. Summary Laconic, Efficient Prover, Laconic, Efficient Prover, HVSZK ARGUMENT Public Key + Encryption CRYPTO HARD LANGUAGE

  14. Techniques

  15. Warmup: 2-Msg, Deterministic Prover * V * a.k.a Hash Proof System [Cramer-Shoup02]

  16. Weak Key Agreement Correctness: Every verifier challenge has Every verifier challenge has unique prover response

  17. D Break average-case hardness V V 0/1 0/1 Adv = Cheating Prover Adv = Cheating Prover soundness soundness Contradiction. D breaks average-case hardness. Amplify from weak PKE to PKE using HolensteinRenner05

  18. We saw: PKE from deterministic, 2-msg SZK Proof System. Challenges: Randomized Prover Multi-round Proof System Stateful Prover Lesser Challenges: Relaxing perfect ZK, perfect completeness Lesser Challenges: Relaxing perfect ZK, perfect completeness

  19. Coping with Randomized Provers Correctness: Weak Security: Weak Security: Trapdoor Our Pseudoentropy PKE Assumption Generator

  20. Trapdoor Our Pseudoentropy PKE Assumption Generator Security : Adv can only sample from “bigger” dist. Formalized using pseudoentropy [HILL99]

  21. Trapdoor Our Pseudoentropy PKE Assumption Generator Challenges: Many rounds Stateful Prover [Ostrovsky 91] Terminate at Laconic. Rejection Sampling random round.

  22. Trapdoor Our Pseudoentropy PKE Assumption Generator Amplification Theorem Technically difficult half Uses connections between Pseudorandomness & Unpredictability Ingredients from: OWFs => PRG (HILL99, VadhanZheng12)

  23. Conclusion and Open Problems Laconic, Efficient Prover, HVSZK ARGUMENT Public Key + + Encryption CRYPTO HARD LANGUAGE Big Open Q: Big Open Q: Design new PKE schemes

  24. Thank You!

  25. Trapdoor Pseudoentropy Public Key Encryption Generator Security : Gap between Decode & adversary Formalized using pseudoentropy [HILL99]

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