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for itself. The public key 
is open to all users; the private key 
is a secret which will be used to issue certificates for user pseudonyms and keying materials, as shown below. In this section, we propose the implicit Comparison-based Profile Matching (iCPM) by adopting the oblivious transfer cryptographic technique [40]. We consider users have distinct values for any given attribute. As shown in Fig. 3, the iCPM consists of three main steps. In the first step, 
chooses an involved category 
by setting y-th element to 1 and other elements to 0 in a λ- length vector 
Thus, 
for 
two encrypted vectors 
and its own attribute value 
In the last step, u in the last step 
if 
or 
if 
for 
generates 2λn secret shares from 
by chossing 
and arranges them in a certain structure according to the predicate For every 2λ secret shares with the same index h, similar to the step 2 of the iCPM, 
or 
from the secret shares.
Then, 
where the comparison operator opt is either > or < and 
The predicate contains n number of requirements (i.e., comparisons), each for a distinct a The responder 
receives 
for attributes 
The initiator 
f the n requirements are satisfied, otherwise. Similar to the iCPM, 
but unknown to 
is chosen by When n = 1, the iPPM reduces to the iCPM. The protocol steps are given below.
shares of 
