Our protocol takes two integers decomposed into encrypted bit vectors [a][b] and outputs the greater integer. In this configuration cloud 1 (C1) has the encrypted bit vectors of the integers being compared and cloud 2 (C2) knows the private key. The protocol is as follows in a very concise form. we can say with firm conviction that vector [Y] consist of encrypted zeros at every location except one location which holds the value of encrypted one. This distinct location identifies the first position where vector [a] and [b] differ.
1. C1 then performs a permutation on vector [Y] and sends it to C2. C2 decrypts the vector and informs C1 where the distinct bit is located. By performing reverse permutation C1 knows precisely where the bit flip occurs and the two key bits that must be compared
2.
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C1 then conducts a coin to randomize the order of which the two key bits are send to C2. C2 decrypts the two bits and evaluates which one is greater and informs C1. C1 then decides whether the condition is satisfied.
During the execution of this protocol no information regarding the value of the two integers being compared is ever revealed to C2. Furthermore, the number of communications between C1 and C2 is constant. To perform range queries over encrypted integers in the cloud, we assume the data owner has encrypted and outsourced their database to C1. Integers in particular are encrypted using their bit representation. The lower and upper bound of the query are encrypted in a similar fashion. The comparison protocol is executed for every record. C1 returns the records that have met the conditions to the end