Multiparty Threshold Private Set Intersection Protocol with Low Communication Complexity

Multiparty threshold private set intersection (MP-TPSI) protocol allows <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>n</mi> </math> mutually untrusted parties <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> holding data sets <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> of size <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>m</mi> </math> respectively to jointly compute the intersection <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>I</mi> <mo>=</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>∩</mo> <mo>⋯</mo> <mo>∩</mo> <msub> <mrow> <mi>A</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> over all their private data sets only if the size of intersection is larger than <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>m</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfenced> </math> , while ensuring that no other private information of the data sets other than the intersection is revealed, where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>t</mi> </math> is the threshold. In the MP-TPSI protocol, multiple parties first decide whether the size of the intersection is larger than the threshold <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mi>t</mi> </math> ; then, they compute the intersection if the size of the intersection is larger than the threshold <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>t</mi> </math> . However, the existing MP-TPSI protocols use different forms of evaluation polynomials in the cardinality testing and intersection computing phases, so that parties need to transmit and calculate a large number of evaluation values, which leads to high communication and computational complexity. In addition, the existing MP-TPSI protocols cannot guarantee the security and the correctness of the results, that is, an adversary can know the additional information beyond the intersection, and the elements that are not in the intersection are calculated as the intersection. To solve these issues, based on the threshold fully homomorphic encryption (TFHE) and sparse polynomial interpolation, we propose an MP-TPSI protocol. In the star network topology, the theoretical communication complexity of the proposed MP-TPSI protocol depends on the threshold <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mi>t</mi> </math> and the number of parties <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi>n</mi> </math> , not on the size of set <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12"> <mi>m</mi> </math> . Moreover, the proposed MP-TPSI protocol outperforms other related MP-TPSI protocols in terms of computational and communication overheads. Furthermore, the proposed MP-TPSI protocol tolerates up to <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </math> corrupted parties in the semi-honest model, where no set of colluding parties can learn the input of an honest party in the strictest dishonest majority setting.