# Secure and Efficient Data Retrieval Process based on Hilbert Space Filling Curve

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3 Data owner Trusted party Data outsourcing based on access policy Generates Data encryption Attribute scrambling Public key Private Key Decrypt Data receiver Data query Cloud Check for keyword using Hilbert space filling curve Retrieve encrypted content Yes Check Sufficient Credentials No No permission granted Fig:1 Architecture diagram Decrypt: Upon receiving the requested content in encrypted form, the receiver obtains the plaintext by using decryption algorithm Decrypt node which can be described as a recursive algorithm. For ease of exposition, we present the simplest form of the decryption algorithm Space filling curve base query search A space-filling curve can be used with a space partition method. A high-dimensional space can be divided into different grid cells, which can be in turn further divided into smaller cells until the cell size or the number of interest objects in the cell is small enough. The level of such partition depends on the smallest cell size and the number of grid nodes in space that space-filling curve can pass through. Each cell is labeled by the unique number (called code) that defines cell s position in the order of space-filling curve. The way of labeling determines the order in which the cells are stored in 1-D media. 3.6 Generation of Hilbert curves Due to the self-similarity property of the Hilbert curve s, higher level Hilbert curve can be generated recursively from the lower level curve. The generation can be done with a set of recursive rules. The 3-D Hilbert curve maps 3-D points into a 1-D linear order. The sequential number (position) of a point on the curve is called Hilbert code. Each Hilbert code has the corresponding enclosing cell in 3-D space. Data points are ordered according to the sequence in which the curve visits the cells that enclose the data points. Each cell can be assigned to base-4 digit ([0-3]) in 2-D and base-8 digit ([0~7]) in 3-D to represent its position relative to the parent (next lower level) cell. The 3-D Hilbert code is a string of base-8 digits, the length of string equals to the coding level. In the encoding step, all the data points are encoded into the initial encoding level. It should be noted that it is not necessary to encode each index point by exactly following the procedure Encoding procedure 1. Given a point (x,y,z) and the 0-level cube; 2. Find the closest vertex i among all 8 vertices of the cube; 3. Set i as Hilbert code for this level; 4. Apply ith permutation rule to get 8 vertices of the sub-cell which contains the point; 5. Repeat step 2-4 until desired coding level; 6. Return Hilbert code string Decoding procedure 1. Get the 0-level cube; 2. i = 1; 3. Pick up the ith digit from Hilbert code string; 4. Apply ith permutation rule to get the vertices of the subcell; 5. i = i + 1; 6. Repeat step 3-5 until i > length (code); 7. Return the vertices of current cell Indexing procedure 1. Choose initial encoding level; 2. Encode data points with initial level; 3. Count number of data points with the same Hilbert code; 38

5 [3] Sahai A, Waters B. Fuzzy identity-based encryption. In: Cramer R, editor. Advances in cryptology, EUROCRYPT 2005, vol Berlin/ Heidelberg: Springer; p [4] Jin, G. and J. Mellor-Crummey (2005). SFCGen: An Framework for Efficient Generation of Multidimensional Space-filling Curves. ACM Transactions on Mathematical Software 31(1): [5] Bethencourt J, Sahai A, Waters B. Ciphertext- policy attribute-based encryption. In: IEEE symposium on security and privacy; p [6] Boneh D, Di Crescenzo G, Ostrovsky R, Persiano G. Public key encryption with keyword search. In: Cachin C, Camenisch J, editors. Advances in Cryptology, EUROCRYPT 2004, vol Berlin/Heidelberg: Springer; p [7] Frikken K, Atallah M, Li J. Attribute-based access control with hidden policies and hidden credentials. IEEE Trans Comput 2006;55: [8] Nishide T, Yoneyama K, Ohta K. Attribute-based encryption with partially hidden encryptor-specified access structures. In: Bellovin S, Gennaro R, Keromytis A, Yung M, editors. Applied cryptography and network security, vol Berlin/Heidelberg: Springer; p IJCA TM : 40

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