1. Introduction to Modern Cryptography Sharif University Spring 2015 Data and Network Security Lab Sharif University of Technology Department of Computer Engineering Sample Applications of Computational Number Theory in Cryptography Author & Instructor: Mohammad Sadeq Dousti 1 / 18

2. Introduction to Modern Cryptography Sharif University Spring 2015  These set of slides are licensed under Creative Commons Attribution-NonCommercial- ShareAlike (CC BY-NC-SA) 4.0.  Basically, this license allows others to use the slides verbatim, and even modify and incorporate them into their own work, as long as: 1. They credit the original author(s); 2. Their work is used non-commercially; 3. They license their work under CC BY-NC-SA 4.0.  For further information, please consult: o https://creativecommons.org/licenses/by-nc-sa/4.0 https://creativecommons.org/licenses/by-nc-sa/4.0 o https://creativecommons.org/licenses/by-nc- sa/4.0/legalcode https://creativecommons.org/licenses/by-nc- sa/4.0/legalcode Copyright Notice 2 / 18

3. Introduction to Modern Cryptography Sharif University Spring 2015  Applications to RSA  RSA is not a secure encryption  Goldwasser–Micali Cryptosystem  Commitment schemes  Coin flipping over the phone  Oblivious transfer  Applications to CFF/CFPs Outline 3 / 18

4. Introduction to Modern Cryptography Sharif University Spring 2015 Applications to RSA 4 / 18

5. Introduction to Modern Cryptography Sharif University Spring 2015 Applications to RSA (Cont’d) 5 / 18

6. Introduction to Modern Cryptography Sharif University Spring 2015 RSA leaks partial information about the message 6 / 18

7. Introduction to Modern Cryptography Sharif University Spring 2015  GM is an encryption scheme. o We will define encryption schemes later. o Informally, the do not leak even partial information.  GM uses a Blum integer n as public key.  The private keys are the factorization of n.  GM encrypts one bit at a time. Goldwasser–Micali Cryptosystem 7 / 18

8. Introduction to Modern Cryptography Sharif University Spring 2015 Goldwasser–Micali Cryptosystem (Cont’d) 8 / 18

9. Introduction to Modern Cryptography Sharif University Spring 2015  A commitment scheme is a security protocol between two parties S (sender) and R (receiver), which has two phases: o Commit o Decommit (or reveal)  Informally: o In the commit phase, S sends a secret value  to R, in such a way that R learns nothing about . o In the decommit phase, S reveals the secret value  to R. - In this phase, it is required that S cannot change the value he committed to. Commitment schemes 9 / 18

10. Introduction to Modern Cryptography Sharif University Spring 2015 1. S generates a random Blum integer N. 2. S encrypts his secret: c = GM(N,  ). 3. Commitment: S sends (N, c) to R. 4. Decommitment: S reveals the randomness used in GM(N,  ) to R.  The above approach works as long as S acts honestly.  What if S chooses N as a non-Blum integer? Using GM to construct a commitment S should prove to R that he picked N honestly. The proof must leak nothing about the factors of N to R. The idea behind zero-knowledge proofs: Proofs that leak nothing but the validity of the statement being proven. S should prove to R that he picked N honestly. The proof must leak nothing about the factors of N to R. The idea behind zero-knowledge proofs: Proofs that leak nothing but the validity of the statement being proven. 10 / 18

11. Introduction to Modern Cryptography Sharif University Spring 2015 Alternative to ZK proofs: Sending decomposition of N 1. S generates a random Blum integer N. 2. S encrypts his secret: c = GM(N,  ). 3. Commitment: S sends (N, c) to R. 4. Decommitment: S sends the factors of N to R. “R verifies that N is a Blum integer, and decrypts .” 1. S generates a random Blum integer N. 2. S encrypts his secret: c = GM(N,  ). 3. Commitment: S sends (N, c) to R. 4. Decommitment: S sends the factors of N to R. “R verifies that N is a Blum integer, and decrypts .” 11 / 18

12. Introduction to Modern Cryptography Sharif University Spring 2015  Assignment 1: Prove that it is a computational binding (assuming log g h modulo p is unknown) and perfect hiding commitment.  Assignment 2: Argue that simultaneous perfect binding & perfect hiding are impossible in commitments. A computational binding & perfect hiding commitment 12 / 18

13. Introduction to Modern Cryptography Sharif University Spring 2015 Coin flipping over the phone 13 / 18

14. Introduction to Modern Cryptography Sharif University Spring 2015  Application 1: Exchange of secrets o Two parties want to exchange their secrets. o Neither is willing to reveal his secret before the other one does so.  Application 2: Contract signing o Two parties want to sign a contract. o Neither is willing to sign before the other one does so.  These applications gave rise to two flavors of OT.  They were shown to be “equivalent.”  OT is sufficiently strong to enable any two-party protocol to be performed. Oblivious Transfer (OT) 14 / 18

15. Introduction to Modern Cryptography Sharif University Spring 2015 Flavor 1:  Sender S has a secret message m.  He wants to “obliviously” transfer it to receiver R. o R receives m with probability ½. o S cannot guess whether R received m or not. Flavor 1:  Sender S has a secret message m.  He wants to “obliviously” transfer it to receiver R. o R receives m with probability ½. o S cannot guess whether R received m or not. OT Flavors 15 / 18

16. Introduction to Modern Cryptography Sharif University Spring 2015  The above protocol requires ZK proofs to be secure against both cheating senders and receivers.  Assignment: Describe how S can cheat. Do the same for R.  We omit an example for OT Flavor 2 for conciseness. OT Flavor 1 16 / 18

17. Introduction to Modern Cryptography Sharif University Spring 2015 Constructing CFF/CFP based on factoring 17 / 18

18. Introduction to Modern Cryptography Sharif University Spring 2015  [Gol01] O. Goldreich. Foundations of Cryptography Volume 1: Basic Tools. Cambridge University Press, 2001.  [KL08] J. Katz and Y. Lindell. Introduction to Modern Cryptography: Principles and Protocols. CRC Press, 2007.  [GB08] S. Goldwasser and M. Bellare. Lecture Notes on Cryptography. 2008. References 18 / 18