Minutiae Matching with Privacy Protection Based on the Combination of Garbled Circuit and Homomorphic Encryption

Algorithm 2

City-block-distance (𝐸𝑀^𝑇, 𝑀^𝑄).

Input: the encrypted template

𝐸𝑀^𝑇=𝑐𝑝₁=𝑚𝑝₁‖⋯‖𝑚𝑝_𝑁𝑃,…,𝑐𝑝_𝑁𝑀=𝑚𝑝_{(𝑁𝑀−1)𝑁𝑃+1}‖⋯‖𝑚𝑝_𝑁𝑇

from Bob, and query minutiae from Alice 𝑀^𝑄

Output: Alice gets a blind list {𝑏𝑥(𝑖,𝑗),𝑏𝑦(𝑖,𝑗),bo(𝑖,𝑗)∣1≤𝑖≤𝑁_𝑇,1≤𝑗≤𝑁_𝑄}

between 𝑀_𝑖=(𝑥_𝑖,𝑦_𝑖,𝜃_𝑖) and 𝑀^′_𝑗=(𝑥^′_𝑗,𝑦^′_𝑗,𝜃^′_𝑗).

(1)   Bob chooses 𝑟^𝑖₁,𝑟^𝑖₂∈_𝑅{0,1}^𝜌+𝛿,𝑟^𝑖₃∈_𝑅{0,1}^𝜇+𝛿, 1≤𝑖≤𝑁_𝑇, concatenates them as strings:

  𝑅₁=𝑟𝑡₁‖⋯‖𝑟𝑡_𝑁𝑃,⋯,𝑅_𝑁𝑀=𝑟𝑡_{(𝑁𝑀−1)𝑁𝑃+1}‖⋯‖𝑟𝑡_𝑁𝑇,

  where 𝑟𝑡_𝑖=0‖𝑟^𝑖₁‖0‖𝑟^𝑖₂‖0‖𝑟^𝑖₃. For 𝑘=1, …, 𝑁_𝑀,

   he computes [𝐴_𝑘]=𝑐𝑝_𝑘[𝑅_𝑘], where 𝐴₁=𝑎¹₁‖𝑏¹₂‖𝑒¹₅‖⋯‖𝑎^𝑁𝑃₁‖𝑏^𝑁𝑃₂‖𝑒^𝑁𝑃₅,

  and so forth and 𝑎_𝑖=𝑟^𝑖₁+𝑥_𝑖, 𝑏_𝑖=𝑟^𝑖₂+𝑦_𝑖, 𝑒_𝑖=𝑟^𝑖₆+𝜃_𝑖. He sends these cipher texts to Alice.

(2)   Alice decrypts [𝐴₁], …, [𝐴_𝑁𝑀] and obtains 𝑎_𝑖=𝑟^𝑖₁+𝑥_𝑖,𝑏_𝑖=𝑟^𝑖₂+𝑥_𝑖 and 𝑒_𝑖=𝑟^𝑖₆+𝜃_𝑖(1≤𝑖≤𝑁_𝑇)

  by parsing 𝐴₁,⋯,𝐴_𝑁𝑀 into 𝜌+𝛿+1-, 𝜌+𝛿+1- and 𝜇+𝛿+1-bit chunks respectively.

  For 𝑖=1, …, 𝑁_𝑇 and 𝑗=1, …, 𝑁_𝑄, she computes 𝑏𝑥(𝑖,𝑗)=𝑎_𝑖−𝑥^′_𝑗=𝑟¹_𝑖+𝑥_𝑖−𝑥^′_𝑗,

  𝑏𝑦(𝑖,𝑗)=𝑏_𝑖−𝑦^′_𝑗=𝑟²_𝑖+𝑦_𝑖−𝑦^′_𝑗

  and bo(𝑖,𝑗)=𝑒_𝑖−𝜃^′_𝑗=𝑟⁶_𝑖+𝜃_𝑖−𝜃^′_𝑗.