Secure integer comparison has been a popular research topic in cryptography,
both for its simplicity to describe and for its applications. The aim is to
enable two parties to compare their inputs without revealing the exact value of
those inputs.

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In this paper, we highlight three-party integer comparison (TPIC), where a
emph{judge}, with no private input, wants to know the comparison result, while
two emph{competitors} hold secret integers to do privacy-preserving
comparison. The judge actively obtains the result rather than passively waiting
for it sent by a competitor. We give two TPIC constructions considering
emph{Mixed adversaries}, who have with different capabilities. One is secure
against a semi-honest adversary with low computation and communication cost,
while the other is secure against a malicious adversary.

Basing on TPIC, we present multi-party comparisons through concrete
applications, including a joint bidding scheme and a practical auction. Brief
security proofs and analysis for the applications are presented. In comparison,
our auction scheme is more efficient with lower cost, making it feasible in
practice rather than a theoretical design. All the comparisons and application
schemes run on top of blockchain requiring a constant number of rounds.

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