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.
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.
Secure multi-party computation is a hot research topic in cryptography in recent years, and has become an important research direction in cryptography. This paper studies the millionaires’ problem in secure multi-party computation, and its essence is a comparison issue about the size of two encrypted data. Yao offered a solution to the millionaires’ problem, but his protocol can’t accurately compare two numbers. We improved Yao’s protocol and proposed two generic protocols to solve the millionaires’ problem. Our first protocol has one more round of communication between participants than Yao’s, but our protocol can accurately compare the size of two numbers. The second protocol improved the first protocol, and only one round of communication is needed to get accurate comparison results. Finally, we realized the second protocol based on RSA encryption algorithm, and analyze the security under the semi-honest model.
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.
The so-called Yao's millionaire problem (YMP) is about securely comparing two numbers held by two parties, respectively. After comparison, the actual values of these two numbers are still unknown to both parties. Recently, Kaghazgaran and Sadeghyan propose a variant of YMP, the two-party multiple comparison (TPMC) protocol, that determines the greatest one over multiple (more than two) encrypted data between two parties. However, in Kaghazgaran and Sadeghyan's TPMC (KS-TPMC) protocol, one of two parties may guess the original data with a probability. In this paper, we introduce a new TPMC protocol to enhance the security by reducing this guess probability, and meanwhile has the same computation cost compared with KS-TPMC protocol. Additionally, the proposed TPMC protocol is more effective on reducing the attack probability for the large number of encrypted data. Thus, our TPMC protocol is suitable for applications on multiple comparison over encrypted data.
Secure comparison problem, also known as Yao's Millionaires' problem, was introduced by Andrew Yao in 1982. It is a fundamental problem in secure multi-party computation. In this problem, two millionaires are interested in determining the richer one between them without revealing their actual wealth. Yao's millionaires' problem is a classic and fundamental problem in cryptography. The design of secure and efficient solutions to this problem provides effective building blocks for secure multi-party computation. However, only a few of the solutions in the literature have succeeded in resisting attacks of malicious adversaries, and none of these solutions has been proven secure in malicious model under ideal/real simulation paradigm. In this paper, we propose two secure solutions to Yao's millionaires' problem in the malicious model. One solution has full simulation security, and the other solution achieves onesided simulation security. Both protocols are only based on symmetric cryptography. Experimental results indicate that our protocols can securely solve Yao's millionaires' problem with high efficiency and scalability. Furthermore, our solutions show better performance than the state-of-the-art solutions in terms of complexity and security. Specifically, our solutions only require O(|U|) symmetric operations at most to achieve simulation-based security against malicious adversaries, where U denotes the universal set and |U| denotes the size of U.
We present three simple and efficient protocol constructions to solve Yao's Millionaire Problem when the parties involved are non-colluding and semi-honest. The first construction uses a partially homomorphic Encryption Scheme and is a 4-round scheme using 2 encryptions, 2 homomorphic circuit evaluations (subtraction and XOR) and a single decryption. The second construction uses an untrusted third party and achieves a communication overhead linear in input bit-size with the help of an order preserving function.Moreover, the second construction does not require an apriori input bound and can work on inputs of different bit-sizes. The third construction does not use a third party and, even though, it has a quadratic communication overhead, it is a fairly simple construction.
In this paper, we present a protocol for generalized secure comparison problem. Yao's millionaire problem is for comparing the wealth of two parties by preserving their privacy. Ifnparties want to compare their wealth by using Yao's protocol, it should be run$n(n-1)2times;iftheirwealthintervalisaslargeasN,thecomplexityofYao′sprotocolfornpartiesisinorderofN.\mathbf{n}^{2}, \mathbf{i}.\mathbf{e}.\mathbf{O}\ (\mathbf{N}.\mathbf{n}^{2})$. In this paper, we present a generalized secure comparison protocol, which need not to be run by every two parties separately. We use an easy knapsack problem for distinguishing parties' indices. Our protocol computation and communication complexities are bounded by 0 (n).