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Lecture notes in networks and systems

A Generalization of Secure Comparison Protocol with Encrypted Output and Its Efficiency Estimation

Takumi Kobayashi, Keisuke Hakuta

Published Sep 16, 2021• 0 citations
Publisher
DOI
10.1007/978-981-16-2380-6_33

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Yao's classical millionaires' problem is about securely determining whether x>y, given two input values x, y, which are held as private inputs by two parties, respectively. The output which is result of comparison becomes known to both parties. In this paper we consider a variant of comparison problem in which the inputs x, y are encrypted and the actual values are not known to the parties. Our first solution deals with single comparison; however, in many applications, we encounter situations where it is necessary to make multiple comparisons to find the maximum among several encrypted data, so we make a modification to our first protocol to solve the multiple comparisons problem. Such a secure comparison is an important building block for applications like privacy preserving data mining and secure business. Our second protocol is based on Homomorphic encryption which achieves very low communication and computational complexities. Also our protocols can be performed in constant rounds and do not use general circuit evaluation techniques so they are more efficient than circuit based ones but not general. Implementation is easy and fast.
Private integer comparison has been an essential computation function for many applications, including online auction, credential identification, data mining, and joint bidding. In the setting of two-party computation, two parties with private inputs (x and y) want to jointly compare them without revealing the value of those inputs to others (also known as the Millionaires’ problem) while the output should ensure correctness and preserve data privacy. The private inputs only can be revealed if they are equal, i.e., x = y. Many related works have been proposed to solve the integer comparison problem in various settings, focusing on different properties such as round and computation complexity. Most solutions decompose integers into bitwise representation and then securely evaluate the function in a Boolean circuit on encrypted bits. However, this type of solution is costly (especially for large integers) as each bit requires encryption and decryption. In this paper, we transform the private integer comparison into a block comparison problem. In particular, we employ a block vectorization mechanism to encode the private inputs into blocks. We show the security of our two-party protocol in the semi-honest model. Also, we implement the protocol to demonstrate its efficiency using block vectorization mechanism and homomorphic encryption. The experimental result proves that our proposed solution achieves high efficiency, particularly for large integer comparisons.
Many privacy protections in distributed setting are based on secure comparison, according to which two distrusted users try to jointly determine whether a > b, a = b, or a1)/(kb + k1)), which implies the potential relationship (a > b, a = b, and a1, and b belong to the party who doesn't have the secret of the system. Based on this system, two distrusted users in distributed setting can learn whether a > b, a = b, or a <; b in one execution, without disclosing a and b to each other. Then, we develop two efficient protocols for secure comparison on integers and secure comparison on fractions, respectively. Our schemes, based on homomorphic encryption, are cryptographically secure. We prove that these protocols are secure using simulation paradigm. Our approaches can be used in many secure multi-party computation protocols that involve fractions, rational numbers, and integers. They can also be more convenient to solve some secure multiparty computational geometry problems that often involve ratio evaluation.