Public Key Encryption That Allows PIR Queries

1. Public Key Encryption That Allows PIR Queries Dan Boneh, Eyal Kushilevitz, Rafail Ostrovsky, William E. Skeith III Presenter:紀汶承

2. Outline • Introduction • Tools • Definition • Main Construction

3. Introduction • Tools • Definition • Main Construction

4. PIR(Private Information Retrieval) • 允許user從擁有database的server中取回資料(item)，但不洩漏取回的是什麼資料。 • PIR solutions • 藉由address，從database中，取回一個 (明文 或 加密過) 的record。 • 靠關鍵字搜尋一個未加密的資料(data)

5. Introduction • Tools • Definition • Main Construction

6. Bloom filter • 觀念: • Hash function: • suppose • A array: such that and such that • Note that ,then

7. Bloom filter(cont.) Input a to hi, i: 1~k h1 h2 hk T 驗證: H2(a) If then H1(a) Hk(a)

8. Bloom filter(cont.) • 儲存什麼? • 不只是單單儲存element，改儲存 : 表示與elements 的關係(表達element所存放的位址) • 現今儲存(a,v), , where • v 被加入 for all • If a∈S,

9. Bloom filter(cont.) Insert: (a1,v1) Insert: (a2,v2) {v1,v2} H1(a1) B1 ∩ B2 {v1} H2(a1) ∩ B3 B4 {v1,v2,v3} ∩ ={v1} Hk(a1) Bm

10. Modifying Encrypted Data in a Communication Efficient Way • Based on group homomorphic encryption with communication O(√n). • Technique : • : database (not encrypted) • (i*,j*): the position of particular element • α: the value we want to add. • v , w: two vector of length √n where • Here δkl = 1 when k=l and 0 otherwise • Then

11. Modifying Encrypted Data in a Communication Efficient Way (cont.) • Parameters: • (K, , D): a CPA-secure public-key encryption • : an array of ciphertexts which is held by a party S. • Define F(X, Y, Z)=X+YZ. By ourassumption, there exists some such that

12. Modifying Encrypted Data in a Communication Efficient Way (cont.) • Protocol: ModifyU,S(l, α) where l and α are private input to U. • U compute i*, j* as the coordinates of l (i.e., i* and j* are quotient and remainder of l/n, respectively) • U sends to S where all values are encrypted under Apublic. • S computes for all , and replaces each cij with the corresponding resulting ciphertext. 每一次修改都對所有的Cij作修改，因此，可以簡易看出保有私密性

13. Introduction • Tools • Definition • Main Construction

14. Definition • 參數: • X: message sending parties. • Y: message receiving party. • S: server/storage provider • 定義: • KeyGen(1S): 產生公密鑰對 • SendX,S(M, K, Apublic) • RetrieveY,S(w, Aprivate)

15. Introduction • Tools • Definition • Main Construction

16. Main Construction • S maintains in its storage space encryptions of the buffers, denote these encryptions • For , we defined • KeyGen(k) :Run K(1s), generate Apublic and Aprivate.

17. SendX,S(M, K, Apublic) ε(M) Server/Storage Sender ρ γcopies of the address ρ ρ ρ ε(M) M + K ρ ρ ρ ρ Message buffer Bloom filter buffer

18. RetrieveY,S(w, Aprivate) PIR query Server/Storage Receiver PIR query, L ε(M) 解密 Message buffer Bloom filter buffer