Homomorphic Encryption Prepared by: Walid A. Hanafy Under - PowerPoint PPT Presentation

Homomorphic Encryption Prepared by: Walid A. Hanafy Under supervision of Professor Mohamed E. Mahmoud

Cryptographic Goals • The Goal is to allow the computations on the encrypted data. • i.e. E 𝑛 � ⊙ 𝑛 � � 𝐹 𝑛 � ⊙ 𝐹�𝑛 � � • Homomorphic Encryption is classified into 3 categories: • Partially Homomorphic: (Only one operation, for unlimited number of executions) • Somewhat Homomorphic: (Multiple operations for a limited number of executions) 2 • Fully Homomorphic: (Multiple operations for an unlimited number of executions)

Partially Homomorphic • Multiplicatively Homomorphic • RSA • El Gamal • Additively Homomorphic • Paillier* * The Paillier Cryptosystem can execute multiplication if only 𝑛 � is encrypted 3

Properties of Paillier Crypto • Depends on hardness of Factorization Problem and The composite residuosity problem. • Encrypted messages are unlinkable 4

Applications • Computation offloading • Secure Data Aggregation: • Smart Metering Infrastructure privacy preservation • E‐Voting 5

How It Works 6

Correctness Proof of Paillier Cryptosystem 7

Preliminaries(1/4): 8

Preliminaries(2/4): 9

Preliminaries(3/4): 10

Preliminaries(4/4): 11

Decryption Phase (1/2) 12

Decryption Phase (2/2) 13

Homomorphism Properties 14

Paper under review • In this paper the homomorphic is applied in three methods: • Spatial Aggregation • Temporal Aggregation • Spatio‐Temporal Aggregation • However, this paper added to the basic HM scheme a threshold condition (Threshold Decryption) 15

Aggregating Spatial Reading (1/2) • For a set of smart meters 𝑡𝑛 � 𝑡𝑛 � , 𝑡𝑛 � , . . , 𝑡𝑛 � , For every interval 𝑞 each meter generates n‐1 random numbers, then each sm computes the following • Then for encryption: where h is the hashed version of 𝑞 . 16

Aggregating Spatial Reading (2/2) • Aggregation: • Given that: 17

Aggregating Temporal Reading • Random Number generation: • Coping with Malfunctions using a third party: 18

Spatio‐Temporal 19

Performance Analysis of Three schemes • Note this scheme is collusion safe as long as colluding parties is less than N‐2. 20