1. Introduction to Modern Cryptography Sharif University Spring 2015 Data and Network Security Lab Sharif University of Technology Department of Computer Engineering Simulation-based Definitions Author & Instructor: Mohammad Sadeq Dousti 1 / 31

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3. Introduction to Modern Cryptography Sharif University Spring 2015  Simulation-based vs. Game-based Definitions o Semantic Security  Zero-Knowledge (ZK) Proofs o Interactive Protocols o Interactive Proof Systems (IPS) o ZK Proofs Outline 3 / 31

4. Introduction to Modern Cryptography Sharif University Spring 2015 Simulation-based vs. Game- based Definitions 4 / 31

5. Introduction to Modern Cryptography Sharif University Spring 2015  Semantics is often stated in one of two flavors: o Game based o Simulation based  Game-based definitions are discussed in the first half of this course.  We will pertain to simulation-based definitions in the second half of the course. Remember: A slide from the beginning of the course 5 / 31

6. Introduction to Modern Cryptography Sharif University Spring 2015  The security model describes the abilities of the adversary.  A game is defined based on these abilities.  The advantage of the adversary is then defined.  Security definition: The scheme is secure if the advantage of any PPT adversary is negligible. Game-based definitions: A template 6 / 31

7. Introduction to Modern Cryptography Sharif University Spring 2015  Threat model: The adversary can eavesdrop on the encryption of a single message.  The model allows the adversary to confine the message space to two messages of her choice. o The messages should be of equal length. o The length of message cannot be hidden. o Proof: Exercise 3.2 of [KL14, p. 102].  The adversary has no knowledge whatsoever about the key. Example: IND-CPA security for private key encryption 7 / 31

8. Introduction to Modern Cryptography Sharif University Spring 2015  Game:  Advantage: Example: IND-CPA security for private key encryption [KL14, p.54] 8 / 31

9. Introduction to Modern Cryptography Sharif University Spring 2015  In defining secure encryption, we reached at the following intuition: Any PPT adversary, given the ciphertext, should not learn anything about the plaintext.  Equivalently: Any information which can be extracted by a PPT adversary about the plaintext given the ciphertext, can be extracted without the ciphertext.  The above notion is called “semantic security.”  How to formalize the semantic security? o Simulation-based definitions! Simulation-based definitions: Motivation 9 / 31

10. Introduction to Modern Cryptography Sharif University Spring 2015  Two security models are defined o A real model: Describes the abilities of the adversary in the real-life world. o An ideal model: Describes the abilities of the adversary in a very idealized model, where the adversary can do almost nothing! Here, the scheme is trivially secure.  Two games are defined based on each model.  The advantage of the adversary in each game is defined.  Security definition: The scheme Π is secure if for any PPT adversary in the real model, there exists a PPT adversary in the ideal model, such that: Simulation-based definitions: A template is often called the simulator, as it should simulate the actions of in the ideal world. 10 / 31

11. Introduction to Modern Cryptography Sharif University Spring 2015  In each game, the advantage of or is defined as the probability that the game outputs 1. Example: Semantic security for private key encryption Samp( ⋅ ) models a distribution over the message space. h models adversary’s a priori information about m. f models any information about m which might leak. 11 / 31

12. Introduction to Modern Cryptography Sharif University Spring 2015  A private-key encryption scheme (E, D) is semantically secure in the presence of an eavesdropper if for every PPT adversary there exists a PPT adversary such that for any PPT algorithm Samp, every poly-time computable functions f and h, every positive polynomial p and all sufficiently large n: Example: Semantic security for private key encryption 12 / 31

13. Introduction to Modern Cryptography Sharif University Spring 2015  Let’s write the definition succinctly as follows:  There are 8 quantifiers!  Try to internalize: o Why each quantifier is needed. o Why the order of quantifiers is stated as above. Semantic security: How many quantifiers? 13 / 31

14. Introduction to Modern Cryptography Sharif University Spring 2015  Semantic security was the first definition proposed for secure encryption. o This is natural, as sematic security closely mimics the perfect security of Shannon with respect to PPT adversaries.  In general, simulation-based definitions are extremely more powerful than game-based ones. o Therefore, any semantically secure encryption is IND-CPA, too.  However, secure encryption is an exception. o It is proven that any IND-CPA encryption is semantically secure, too.  When working with secure encryptions: o Semantic security is more naturally described. o IND-CPA encryption is easier to work with. A bit more explanation 14 / 31

15. Introduction to Modern Cryptography Sharif University Spring 2015 Zero-Knowledge (ZK) 15 / 31

16. Introduction to Modern Cryptography Sharif University Spring 2015  In many cryptographic situations, it is desirable to prove a fact, without revealing further knowledge.  Example: o In some cryptographic protocol, party P should choose a random Blum integer n, yet keep the factors of n secret. o The other party V is reluctant whether P acted honestly. o P should prove to V his honesty, without revealing any knowledge about the factors of n. o A zero-knowledge proof is the solution!  ZK is invented by Goldwasser, Micali, and Rackoff (GMR) in 1985.  The ZK definition is simulation based. The idea behind ZK 16 / 31

17. Introduction to Modern Cryptography Sharif University Spring 2015  GMR generalized the notion of proofs.  Classically, a proof is identical to an NP witness. o When you prove something, it is a short witness which can be verified in time polynomial in the size of the problem.  GMR proofs differ from classical proofs in two respects: o Interactivity: The prover (P) and verifier (V) interact for a number of rounds. o Probabilism: Both parties may toss coin. At the end of the proof, the verifier should be convinced with high probability (recall completeness & soundness).  Both interactivity & probabilism are crucial for ZK. A word on the notion of proofs 17 / 31

18. Introduction to Modern Cryptography Sharif University Spring 2015  GMR introduced the notion of interactive Turing machines (ITM).  An ITM is a probabilistic Turing machine, endowed with one read-only (RO) and one write-only (WO) communication tape.  An interactive protocol is two ITMs, paired as follows: o The ITMs share the input tape. Moreover, the RO tape of one ITM becomes the WO tape of another, and vice versa. o The ITMs take turns in computation. o The turn finishes when one machine writes something on its WO tape. o If one ITM halts, the other ITM activates for the last turn. GMR model for ZK 18 / 31

19. Introduction to Modern Cryptography Sharif University Spring 2015 A pair of ITMs P P V V Random tape of PRandom tape of V Common input tape Work tape of PWork tape of V WO tape of P RO tape of V RO tape of P WO tape of V read read/write write 19 / 31

20. Introduction to Modern Cryptography Sharif University Spring 2015  Prover P has unlimited computational power.  Verifier V is PPT.  The common input is x  {0,1} *.  P intends to prove to V that x belongs to some set L.  At the end of interaction, V halts in either accepting or rejecting state. o {V, P}(x) = 1 if V accepts. o {V, P}(x) = 0 if V rejects. Notation 20 / 31

21. Introduction to Modern Cryptography Sharif University Spring 2015 Interactive proofs 21 / 31

22. Introduction to Modern Cryptography Sharif University Spring 2015 Example: Graph isomorphism No PPT algorithm is known to decide whether two graphs are isomorphic. [Wikipedia, user Chris Martin] 22 / 31

23. Introduction to Modern Cryptography Sharif University Spring 2015 Interactive proof for graph isomorphism 23 / 31

24. Introduction to Modern Cryptography Sharif University Spring 2015  Informally, an interactive protocol (V, P)(x) is ZK for x  L if anything that any verifier V * sees during the protocol can be computed without the help of P.  Whatever V * sees during the protocol is called the view of V *. o Common input (x) o Random tape of V * (r) o RO communication tape of V * (incoming messages from P)  View of V * is a random variable, denoted (x). ZK: Informal definition 24 / 31

25. Introduction to Modern Cryptography Sharif University Spring 2015  The real world is modeled as a pair of ITMs (P, V * ), having access to the common input x.  The ideal world is modeled by a probabilistic (expected) polynomial-time algorithm S (the simulator), having access to the common input x. o S cannot interact with P.  As simple as that! o The models are so simple that they are often embedded within the security definition of ZK. ZK: Real and ideal models 25 / 31

26. Introduction to Modern Cryptography Sharif University Spring 2015  The advantage in each game is defined as the probability that D outputs 1.  An interactive protocol (P,V) is called ZK on L if for any PPT adversary V *, there exists a probabilistic (expected) polynomial-time algorithm S, such that for all PPT distinguisher D, any positive polynomial q, and all sufficiently large x  L: ZK: Real and ideal games Like other simulation- based definitions, S can depend on the adversary V *. 26 / 31

27. Introduction to Modern Cryptography Sharif University Spring 2015  The view of V * is: o Common input x = (G 0, G 1 ). o A random tape (r). o Messages received from the prover: (H, f ) or (H, ⊥ ).  How can the simulator succeed?! o The simulator can depend on V *. o It can run the code of V * internally (as a subroutine). o It can re-run and try various inputs for V *. Simulating the IPS for GI 27 / 31

28. Introduction to Modern Cryptography Sharif University Spring 2015 Simulating the IPS for GI (Cont’d) 28 / 31 Assignment: (1)Prove that S runs in expected polynomial time. (2)Prove that the output of S and the view of V * are identically distributed. Assignment: (1)Prove that S runs in expected polynomial time. (2)Prove that the output of S and the view of V * are identically distributed.

29. Introduction to Modern Cryptography Sharif University Spring 2015  THEOREM 1: Anything provable can be proven in zero-knowledge. o In other words, every IPS can be re-written as an IPS with ZK property. o In particular, any NP statement has a ZK proof.  THEOREM 2: Interaction and randomness are essential to have ZK proofs for non-trivial languages [GO94, GK96]. o However, non-interactive ZK (NIZK) is possible if we change the model [FLS99]. Two theorems 29 / 31

30. Introduction to Modern Cryptography Sharif University Spring 2015 Some observations 30 / 31

31. Introduction to Modern Cryptography Sharif University Spring 2015  [FLS99] U. Feige, D. Lapidot, and A. Shamir. Multiple Noninteractive Zero Knowledge Proofs Under General Assumptions, SIAM J. COMPUT., 1999.  [GK96] O. Goldreich and H. Krawczyk. On the Composition of Zero-Knowledge Proof Systems, SIAM J. COMPUT., 1996.  [GO94] O. Goldreich and Y. Oren. Definitions and Properties of Zero-Knowledge Proof Systems, J. Cryptology, 1994.  [Gol01] O. Goldreich. Foundations of Cryptography, Volume 1: Basic Tools. Cambridge University Press, 2001.  [Gol04] O. Goldreich. Foundations of Cryptography, Volume 2: Basic Applications. Cambridge University Press, 2004.  [KL14] J. Katz and Y. Lindell. Introduction to Modern Cryptography: Principles and Protocols. 2 nd Edition, CRC Press, 2014. References 31 / 31