2. Definition and applications Lossy Trapdoor Functions 2

3. Definition [PW08] 3 Invertible Lossy

4. Lossy Trapdoor Functions Implications 4 LTDF [PW08] TDF IND-CPA Det. Enc.[BFO08] (New!) Hedged Enc.[BB+09] (New!) Others… [BKPW12] What about the IB setting?

5. Lossy Trapdoor Functions Constructing a primitive 5 SetupEncryptDecrypt Gen Eval pk Invert SetupEncryptDecrypt? Gen’ Eval pk’ Game 1 Game 2 Secure! C hides M! IND

6. Working towards a definition Identity-Based Lossy Trapdoor Function [BKPW12] 6

7. Identity-Based Lossy Trapdoor Functions IBE - Functionality 7 Constructed with an IB-LTDF uses: IBE [Sha84,BF01] consists of:

8. Identity-Based Lossy Trapdoor Functions Functional requirements 8 Invertible

9. Identity-Based Lossy Trapdoor Functions IBE – Security Game / Reduction 9 can try to invert. should be lossy Using IB-LTDF

10. Identity-Based Lossy Trapdoor Functions Towards defining sec. requirements 10 Sec. Requirement? Invertible Lossy

11. Identity-Based Lossy Trapdoor Functions [BKPW12] limitations 11 LTDF [PW08] IB-LTDF (S) [BKPW12] TDF (New!) IND-CPA Det. Enc.[BFO08] (New!) Hedged Enc.[BB+09] (New!) Others…hopefully

12. Identity-Based Lossy Trapdoor Functions [BKPW12] limitations 12 LTDF [PW08] IB-LTDF (S) [BKPW12] IB-LTDF (A) [BKPW12] TDF (New!) IND-CPA ? Det. Enc.[BFO08] (New!) ? Hedged Enc.[BB+09] (New!) ? Others…hopefully?

13. New Definition and Hierarchical Extension Identity-Based Lossy Trapdoor Function 13

14. Identity-Based Lossy Trapdoor Functions Our definition (I) 14 Real Experiment I L has small range has full range Lossy Experiment

15. Identity-Based Lossy Trapdoor Functions Our definition (II) 15 Extra Cond. #1: big enough Extra Cond. #2: indep. from guess

16. Identity-Based Lossy Trapdoor Functions [EHLR14] implications 16 LTDF [PW08] IB-LTDF (S) [EHLR14] IB-LTDF (A) [EHLR14] TDF IND-CPA Det. Enc.[BFO08] (New!) (New!)* Hedged Enc. [BB+09] (New!) (New!) Others…hopefully *Also in [XXZ12]

17. Identity-Based Lossy Trapdoor Functions [EHLR14] implications 17 LTDF [PW08] IB-LTDF (S) [EHLR14] IB-LTDF (A) [EHLR14] HIB-LTDF (S,A) [EHLR14] TDF (New!) IND-CPA Det. Enc.[BFO08] (New!) (New!)* (New!) Hedged Enc. [BB+09] (New!) (New!) Others…hopefully Using [CHK03]… [EHLR14]  Forward Secure Det. Enc. (New!) [EHLR14]  Forward Secure Hedged Enc. (New!)

18. Construction Identity-Based Lossy Trapdoor Function 18

19. Identity-Based Lossy Trapdoor Functions Construction similar to [PW08] o Matrix-vector paradigm Building block: a new Hierarchical Predicate Encryption o Hidden Predicate defines Injective or Lossy To evaluate the function for an identity: 1.Homomorphically evaluate the Predicate for the Identity 2.Obtain a matrix of HIBE ciphertexts 3.Compute the matrix-vector product in the exponent Our construction 19

20. Conclusion Identity-Based Lossy Trapdoor Function 20

21. Identity-Based Lossy Trapdoor Functions We give a new definition We give a hierarchical extension of the definition Our definition implies new primitives with adaptive security: o One-way HIB Trapdoor Functions o HIB Deterministic Encryption o HIB Hedged Encryption o Forward Secure Deterministic Encryption o Forward Secure Hedged Encryption We give a construction which satisfies the extended definition Our contributions 21

22. THANK YOU!