Table 3.1: Protocol Efficiency Comparison
Protocol [13] [21] [15] ΠEEQ ΠPEEQ
Preprocessing Phase
Communication O(ℓ2) O(ℓ2) O(ℓ2) ⊥ O(ℓlog(ℓ))
Computation O(ℓ3) O(ℓ3) O(ℓ3) ⊥ O(ℓlog(ℓ))
Rounds O(1) 9 2 ⊥ 1
Online Phase
Communication O(ℓ) O(ℓ) O(ℓ2) O(ℓlog(ℓ)) O(ℓ) Computation O(ℓ2) O(ℓ2) O(ℓ3) O(ℓlog(ℓ)) O(ℓ)
Rounds 2 2 6 2 2
Protocol [16] [15] [21] ΠBLT ΠPBLT
Preprocessing Phase
Communication O(ℓ2) O(ℓ2) O(ℓ2/log(ℓ)) ⊥ O(ℓlog(ℓ) log(log(ℓ))) Computation O(ℓ3) O(ℓ3) O(ℓ3/log(ℓ)) ⊥ O(ℓlog(ℓ) log(log(ℓ)))
Rounds 6 2 3 ⊥ 1
Online Phase
Communication O(ℓ2) O(ℓ2) O(ℓ2/log(ℓ)) O(ℓlog(ℓ) log(log(ℓ))) O(ℓlog(ℓ)) Computation O(ℓ3) O(ℓ3) O(ℓ3/log(ℓ)) O(ℓlog(ℓ) log(log(ℓ))) O(ℓlog(ℓ))
Rounds 3 6 4 3 3
Protocol [15] [19] [17] ΠBD ΠBD′
Overall
Communication O(ℓ2·log(ℓ)) O(c·ℓ·log∗(c)(ℓ)) O(ℓ2) O(ℓ3) O(ℓ2log(ℓ) log(log(ℓ))) Computation O(ℓ3·log(ℓ)) O(c·ℓ2·log∗(c)(ℓ)) O(ℓ3) O(ℓ3) O(ℓ2log(ℓ) log(log(ℓ)))
Rounds (E) 25 (E) 23 +c (E) 12 2 3
(E) Specifies that a protocol only runs in expected constant rounds.
Source: Created by author.
Our results are presented in Table 3.1. We once more emphasize that our comparisons and conclusions are taken with respect to other 2-party protocols present in the literature, with unconditional perfect security.
Our protocol ΠPEEQ is the only 2-party protocol for secure equality test with unconditional, perfect security that communication and computational complexities linear inℓ. ΠEEQ is the only protocol that has 2 rounds in the online phase, without a preprocessing phase -at the cost of increasing the comput-ation and communic-ation complexities toO(ℓlog(ℓ)). Our protocols for private comparison, ΠBLT and ΠPBLT, are the only protocols that have three rounds with communication and computational complexities equal to O(ℓlog(ℓ) log(log(ℓ))).
Finally, our protocols for bit decomposition ΠBDand Π′BDare the only ones in the literature that have three or less rounds.
Chapter 4 Conclusions
Protocols for secure equality tests, comparison, and bit decomposition are important re-search areas within cryptographic protocols, with their relevance becoming clearer because of the crucial role they play in privacy-preserving machine learning protocols [10, 7, 11].
In this work, we have shown how to build novel protocols for all these functions, improving on the efficiency of previous published solutions, specially on the number of communi-cations of rounds betweem the two parties. In order to do this, we make use of a new cryptographic primtive called Shared Oblivious Transfer, which we believe can be of inde-pendent interest, that can be seen as an extension of the widely known Oblivious Transfer primitive.
These new results lead to other interesting questions that are not answered in this work and that could be the focus of future ones. The protocols for equality, comparison and bit-decomposition are prove secure against semi-honest adversaries, that is, adversaries that follow the described protocol. A natural question is if we can modify these protocol to make them secure against adversaries that behave arbitrarily while sacrificing minimal performance.
We saw in this work how Shared OTs can be used to improve the efficiency of protocols for multiple interesting and useful functionalities. Another natural question that this leads to is if this new cryptographic primitive can be applied to improve the efficiency of other interesting functionalities, such as set intersection of secrets and elementwise comparison.
Future efforts could be dedicated to studying other functionalities that could benefit from Shared OT.
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