k-nearest Neighbor Classification over Semantically Secure Encry
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1 k-nearest Neighbor Classification over Semantically Secure Encrypted Relational Data Reporter:Ximeng Liu Supervisor: Rongxing Lu School of EEE, NTU May 9, 2014
2
3 Outline 1. Samanthula B K, Elmehdwi Y, Jiang W. k-nearest Neighbor Classification over Semantically Secure Encrypted Relational Data[J]. arxiv preprint arxiv: , Elmehdwi Y, Samanthula B K, Jiang W. Secure k-nearest Neighbor Query over Encrypted Data in Outsourced Environments[C]. ICDE 2014, accepted.
4 k-nearest Neighbor algorithm
5 Problem Definition Suppose Alice owns a database D of n records t 1,, t n and m + 1 attributes. Let t i,j denote the j-th attribute value of record t i. Initially, Alice encrypts her database attribute-wise, that is, she computes E pk (t i,j ), for 1 i n and 1 j m + 1, where column (m + 1) contains the class labels. Let the encrypted database be denoted by D.
6 Problem Definition Let Bob be an authorized user who wants to classify his input record q =< q 1,, q m > by applying the k-nn classification method based on D. We refer to such a process as privacy-preserving k-nn (PPkNN) classification over encrypted data in the cloud. Formally, we define the PPkNN protocol as: PPkNN(D, q) c q where c q denotes the class label for q after applying k-nn classification method on D and q.
7 Paillier Cryptosystem a. Homomorphic Addition E pk (a + b) = E pk (a) E pk (b) mod N 2 b. Homomorphic Multiplication E pk (a b) = E pk (a) b mod N 2
8 Basic Privacy-Preserving Protocols All of the below protocols are considered under two-party semi-honest setting. In particular, we assume the exist of two semi-honest parties P 1 and P 2 such that the Paillier s secret key sk is known only to P 2 whereas pk is treated as public. 1. Multiplication (SM) Protocol: This protocol considers P 1 with input (E pk (a), E pk (b)) and outputs E pk (a b) to P 1, where a and b are not known to P 1 and P 2. During this process, no information regarding a and b is revealed to P 1 and P 2.
9 Basic Privacy-Preserving Protocols Secure Squared Euclidean Distance (SSED) Protocol: In this protocol, P 1 with input (E pk (X ), E pk (Y )) and P 2 with sk securely compute the encryption of squared Euclidean distance between vectors X and Y. Here X and Y are m dimensional vectors where E pk (X ) =< E pk (x1),, E pk (x m ) > and E pk (Y ) =< E pk (y 1 ),, E pk (y m ) >. The output of the SSED protocol is E pk ( X Y 2 ) which is known only to P 1
10 Basic Privacy-Preserving Protocols Secure Bit-Decomposition (SBD) Protocol: P 1 with input E pk (z) and P 2 securely compute the encryptions of the individual bits of z, where 0 z < 2 l. The output [z] =< E pk (z 1 ),, E pk (z l ) > is known only to P 1. Here z 1 and z l are the most and least significant bits of integer z, respectively
11 Basic Privacy-Preserving Protocols Secure Minimum (SMIN) Protocol: In this protocol, P 1 holds private input (u, v ) and P 2 holds sk, where u = ([u], E pk (s u )) and v = ([v], E pk (s v )). Here s u (resp., s v ) denotes the secret associated with u (resp., v). The goal of SMIN is for P 1 and P 2 to jointly compute the encryptions of the individual bits of minimum number between uand v. In addition, they compute E pk (smin(u, v)). That is, the output is ([min(u, v)], E pk (s min(u,v) )) which will be known only to P 1. During this protocol, no information regarding the contents of u, v, s u, and s v is revealed to P 1 and P 2.
12 Basic Privacy-Preserving Protocols Secure Minimum out of n Numbers (SMIN n ) Protocol: In this protocol, we consider P 1 with n encrypted vectors ([d 1 ],, [d n ]) along with their respective encrypted secrets and P 2 with sk. Here [d i ] = E pk (d i,1 ),, E pk (d i,l ) where d i,1 and d i,l are the most and least significant bits of integer d i respectively, for 1 i n. The secret of d i is given by s di. P 1 and P 2 jointly compute [min(d 1,, d n )]. In addition, they compute E pk (s min(d1,,d n)). At the end of this protocol, the output ([min(d 1,, d n )], E pk (s min(d1,,d n))) is known only to P 1. During the SMIN n protocol, no information regarding any of d i s and their secrets is revealed to P 1 and P 2.
13 Basic Privacy-Preserving Protocols Secure Bit-OR (SBOR) Protocol: P 1 with input (E pk (o 1 ), E pk (o 2 )) and P 2 securely compute E pk (o 1 o 2 ), where o 1 and o 2 are two bits. The output E pk (o 1 o 2 ) is known only to P 1.
14 Secure Multiplication (SM) a b = (a + r a ) (b + r b ) a r a b r b r a r b Note that, for any given x Z N, N x is equivalent to x under Z N
15 Secure Multiplication (SM)
16 Secure Squared Euclidean Distance (SSED)
17 Secure Bit-Decomposition (SBD) We assume that P 1 has E pk (z) and P 2 has sk, where z is not known to both parties and 0 z < 2 l. Given E pk (z), the goal of the secure bit-decomposition (SBD) protocol is to compute the encryptions of the individual bits of binary representation of z. That is, the output is [z] =< E pk (z 1 ),, E pk (z l ) >, where z 1 and z l denote the most and least significant bits of z respectively. At the end, the output [z] is known only to P 1. During this process, neither the value of z nor any z i s is revealed to P 1 and P 2.
18 Secure Minimum (SMIN) the basic idea of the proposed SMIN protocol is for P 1 to randomly choose the functionality F (by flipping a coin), where F is either u > v or v > u, and to obliviously execute F with P 2. Since F is randomly chosen and known only to P 1, the result of the functionality F is oblivious to P 2. Based on the comparison result and chosen F, P 1 computes [min(u, v)] and E pk (s min(u,v) ) locally using homomorphic properties.
19 Secure Minimum (SMIN)
20 Secure Minimum (SMIN)
21 Secure Minimum (SMIN)
22 Secure Minimum (SMIN)
23 Secure Minimum (SMIN)
24 Secure Minimum out of n Numbers (SMIN n )
25 Secure Bit-OR (SBOR) Suppose P 1 holds (E pk (o 1 ), E pk (o 2 )) and P 2 holds sk, where o 1 and o 2 are two bits not known to both parties. The goal of the SBOR protocol is to securely compute E pk (o 1 o 2 ). At the end of this protocol, only P1 knows E pk (o 1 o 2 ). During this process, no information related to o 1 and o 2 is revealed to P 1 and P 2. Given the secure multiplication (SM) protocol, P 1 can compute E pk (o 1 o 2 ) as follows:
26 P 1 with input (E pk (o 1 ), E pk (o 2 )) and P 2 involve in the SM protocol. At the end of this step, the output E pk (o1 o2) is known only to P 1. Note that, since o 1 and o 2 are bits, Epk(o1 o2) = Epk(o1 o2). Epk(o1 o2) = Epk(o1 + o2) Epk(o1 o2) N 1
27 Basic scheme Outline
28 Basic scheme Outline
29 Fully Secure knn Protocol
30 Fully Secure knn Protocol
31 Thank you Rongxing s Homepage: PPT Ximeng s Homepage:
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