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1 Adaptively Attribute-Hiding ( Hierarchical ) Inner Product Encryption 2012 / 4 / 18 Tatsuaki Okamoto ( NTT ), Katsuyuki Takashima ( Mitsubishi Electric ).

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2 Functional Encryption Plain text Encryption Cipher text Public key pk Decryption Plain text Secret key with parameter Parameter sk Relation R(, ) holds This type is called Predicate Encryption in [BSW11].

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3 Inner Product Encryption ( IPE ) [KSW08]

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4 (Adaptive Secure &) Weakly Attribute-Hiding IPE Challenger Some additional information on may be revealed to a person with a matching key, i.e.,

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5 (Adaptive Secure &) Fully Attribute-Hiding IPE Challenger No additional information on is revealed even to any person with a matching key, i.e., For each run of the game, the variable is defined as if otherwise.

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6 [ LOS + 10 ] : Adaptively secure but weakly attribute- hiding IPE based on a non-standard assumption [ KSW08 ] : Fully attribute-hiding but selectively secure IPE Previous works of Attribute-Hiding IPE [ OT10 ] : Adaptively secure but weakly attribute-hiding IPE based on the DLIN assumption [ AFV11 ] : Selectively secure and weakly attribute-hiding IPE based on the LWE assumption Adaptively secure and fully attribute-hiding IPE based on the DLIN assumption This work

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7 Our Results Adaptively secure and fully attribute-hiding IPE based on the DLIN assumption (basic scheme) A variant IPE with a shorter ( O(n) -size) master public key and shorter ( O(1) -size) secret keys (excluding the description of ) An extension to Hierarchical IPE (HIPE) with the same security

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8 Key Techniques Dual Pairing Vector Space (DPVS) approach provides rich basic transformations for achieving these various forms. All forms of a secret-key do not depend on whether it is matching or not. Large ( -dim.) hidden subspaces gives new types (Types 1-3) of information theoretical tricks and various forms of computational reductions. We extend Dual System Encryption (DSE) for our purpose with various forms, i.e., normal, temporal 1, temporal 2 and unbiased …. Fully-AH IPE should deal with both cases, matching and non-matching keys (to challenge CT), while weakly-AH IPE deals with only the non-matching case.

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9 Dual Pairing Vector Space Approach (I) Vector space using symmetric pairing groups whereis a generator of ( Canonical ) pairing operation: For and where dual orthonormal bases of i.e., Dual Bases : basis of for s.t.

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10 DPVS Approach (II) with ( the canonical Cryptographic Construction using Dual Pairing Vector Space (DPVS) approach : pairing and ) random dual bases as a master key pair DLIN-based security from [OT10] machinery For and we denote Notation : Basic Fact for Our Construction For the aboveand from dual orthonormality of

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11 Intractable Problems on DPVS Security of our IPE is proven under DLIN assumption, through variants of DSP. Vector Decomposition Problem (VDP) : Dual Basis Computation Problem (DBP) : Hard to calculate (master secret) from (master public) E.g., hard to calculate from Decisional Subspace Problem (DSP) : Hard to distinguish and where DBP Assump.VDP Assump. DSP Assump. DLIN Assump.

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12 Basic Idea for Constructing IPE using DPVS where

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13 Weakly Attribute-Hiding IPE Scheme in [OT10] where

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14 Proposed (Basic) Fully Attribute-Hiding IPE Scheme where

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15 Game 0 Challenger We define that wins with prob. 1/2 when the game is aborted in Game 0. negligible from [OT10] target of this talk -> Game 0 if otherwise Game 0 is the same as real security game, Game 0, except that flip a coin before setup and the game is aborted if

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16 Dual System Encryption (DSE) Methodology (I) 1)Challenge ciphertext Semi-func. 2)Keys Semi-func. (one by one) 3)Semi-func. challenge ciphertext Random i.e., Advantage of adversary = 0 Simulator can change them under the above conditions. Simulator …

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17 DSE Methodology (II) Normal key Semi-func. key This semi-func. form of keys cannot be used for fully-AH. Need to introduce new forms with preserving functionality Normal ciphertext Semi-func. ciphertext

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18 Extension of DSE (I): R-preserving ciphertexts independent of challenge bit Aim of game transformation: Transform to -unbiased CT, for (all but negligible prob.) I.e., & Independent of bitpreserving

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19 Extension of DSE (II): Randomization in 2-dim. and Swapping Temporal 1 Key with DLIN Temporal 1 CT with DLIN Temporal 2 Key with swapping Temporal 2 CT with randomization Iterate the changes among these 4 forms for all queried for preparing the next

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20 Extension of DSE (III): Last Conceptual Change to Unbiased CT Temporal 2 CT with Temporal 2 Key with 1-st block for randomization 2-nd block for keeping In Game 2- -4, All queried keys are Unbiased CT with which is unbiased of is obtained. In Game 3, is bounded by advantages for DLIN

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21 Original DSE Methodology Comparison of Original and Extension of DSE 1)Challenge CT Semi-func. 2)Keys Semi-func. (one by one) 3)CT Random Extension of DSE 1)Challenge CT 2)Keys CT random since since 3)CT Unbiased w.r.t. b (one by one)

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22 Key Ideas for Short Public / Secret Key IPE We will explain key ideas using -dim. basic IPE. We employ a special form of master secret key basis, where and a blank in the matrix denotes Secret-key associated with can be compressed to only 3 group elements Then, as well as

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23 Special Basis for fully-AH IPE with Short SK We extend the basic construction to a 5 x 5 block matrix one to achieve full AH security (as our basic IPE).

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24 Adaptively Fully-AH IPE with Constant-Size SK SK size

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25 Thank You !

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