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

Algorithm 1

Euclidean-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 the blind squared-distance bd 𝑖,𝑗 and blind direction bo 𝑖,𝑗 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^𝑖₄∈_𝑅{0,1}^2𝜌+𝛿, 𝑟^𝑖₆∈_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‖0‖𝑟^𝑖₄‖0‖𝑟^𝑖₆‖.

   For 𝑘=1, …, 𝑁_𝑀, he computes 𝐴_𝑘=𝑐𝑝_𝑘𝑅_𝑘,

  where 𝐴₁=𝑎¹₁‖𝑏¹₂‖𝑐¹₃‖𝑑¹₄‖𝑒¹₅‖⋯‖𝑎^𝑁𝑃₁‖𝑏^𝑁𝑃₂‖𝑐^𝑁𝑃₃‖𝑑^𝑁𝑃₄‖𝑒^𝑁𝑃₅, and so forth

  and 𝑎_𝑖=𝑟^𝑖₁+𝑥_𝑖, 𝑏_𝑖=𝑟^𝑖₂+𝑦_𝑖, 𝑐_𝑖=𝑟^𝑖₃+𝑥²_𝑖, 𝑑_𝑖=𝑟^𝑖₄+𝑦²_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,𝑐_𝑖=𝑟^𝑖₃+𝑥²_𝑖,𝑑_𝑖=𝑟^𝑖₄+𝑦²_𝑖,

   and 𝑒_𝑖=𝑟^𝑖₆+𝜃_𝑖 (1≤𝑖≤𝑁_𝑇) by parsing 𝐴₁,…,𝐴_𝑁𝑀 into 𝜌+𝛿+1-, 𝜌+𝛿+1-, 2𝜌+𝛿+1-,

   2𝜌+𝛿+1- and 𝜇+𝛿+1-bit chunks respectively. And:

   (i)   For 𝑖=1, …, 𝑁_𝑇 and 𝑗=1, …, 𝑁_𝑄, she calculates 𝑒𝑑_𝑖𝑗=𝑐_𝑖+𝑑_𝑖−2𝑥^′_𝑗𝑎_𝑖−2𝑦^′_𝑗𝑏_𝑖+𝑥^′2_𝑗+𝑦^′2_𝑗.

  (ii)   She pads 𝜆−𝜌−1 zeros before each 2𝑥^′_𝑗 and 2𝑦^′_𝑗,

       constructs the 𝜆-bit strings: 𝑢¹_𝑗=0^{(𝜆−𝜌−1)}‖2𝑥^′_𝑗 and 𝑢²_𝑗=0^{(𝜆−𝜌−1)}‖2𝑦^′_𝑗, where 𝜆=2𝜌+𝛿+3.

       She then concatenates these strings together respectively: 𝐵₁=𝑢¹₁‖⋯‖𝑢¹_𝑁𝑋, …,

      𝐵_𝑁𝐵=𝑢¹_{(𝑁𝐵−1)𝑁𝑋+1}‖⋯‖𝑢¹_𝑁𝑄, 𝐶₁=𝑢²₁‖⋯‖𝑢²_𝑁𝑋, …, 𝐶_𝑁𝐵=𝑢²_{(𝑁𝐵−1)𝑁𝑋+1}‖⋯‖𝑢²_𝑁𝑄,

      where 2𝑥^′_𝑗 and 2𝑦^′_𝑗 are both 𝜌+1-bit values, 𝑢¹_𝑖 and 𝑢²_𝑖 are both 𝜆-bit values, 𝑁_𝑋=⌈𝑇/𝜆⌉

      and 𝑁_𝐵=𝑁_𝑄/𝑁_𝑋.

  (iii)    For 𝑘=1, …, 𝑁_𝐵, she computes 𝐵_𝑘 and 𝐶_𝑘, transfers them to Bob.

(3)  Bob receives the cipher texts. Then:

   (i)   For 𝑖=1,…, 𝑁_𝑇 and 𝑘=1, …, 𝑁_𝐵, he computes 𝑡¹_𝑖,𝑘=𝐵_𝑘^𝑟1𝑖𝐶_𝑘^𝑟2𝑖=𝑟¹_𝑖𝐵_𝑘+𝑟²_𝑖𝐶_𝑘.

  (ii)   He chooses 𝑟⁵_𝑖,𝑗∈_𝑅{0,1}^𝜆−1, 1≤𝑖≤𝑁_𝑄, 1≤𝑗≤𝑁_𝑄, pads a zero before each number,

         concatenates each 𝑁_𝑋 numbers as strings with respect to 𝑗:

         𝑡²_𝑖,1=0‖𝑟⁵_𝑖,1‖⋯‖0‖𝑟⁵_{𝑖,𝑁𝑋}, …, 𝑡²_{𝑖,𝑁𝐵}=0‖𝑟⁵_{𝑁𝑖,𝐵𝑁−1𝑋+1}‖⋯‖0‖𝑟⁵_{𝑖,𝑁𝑄}, and computes

       𝑡²_𝑖,1, …, 𝑡²_{𝑖,𝑁𝐵}.

  (iii)   For 𝑖=1,…, 𝑁_𝑇 and 𝑘=1,…, 𝑁_𝐵, he computes 𝑡³_𝑖,𝑘=𝑡¹_𝑖,𝑘𝑡²_𝑖,𝑘=𝑡¹_𝑖,𝑘+𝑡²_𝑖,𝑘

         and sends them to Alice.

(4)  Alice receives and decrypts 𝑡³_𝑖,𝑘, 1≤𝑖≤𝑁_𝑇, 1≤𝑘≤𝑁_𝐵. Then:

  (i)     For 𝑖=1, …, 𝑁_𝑇, she unpacks each 𝑡³_𝑖,𝑘 by parsing it into 𝜆-bit chunks to obtain

       𝑡⁴_𝑖,1, …, 𝑡⁴_{𝑖,𝑁𝑋}, …, 𝑡⁴_{𝑖,(𝑁𝐵−1)𝑁𝑋+1}, …, 𝑡⁴_{𝑖,𝑁𝑄}, where 𝑡⁴_𝑖,𝑗=2𝑟¹_𝑖𝑥^′_𝑗+2𝑟²_𝑖𝑦^′_𝑗+𝑟⁵_𝑖,𝑗.

  (ii)   For 𝑖=1, …, 𝑁_𝑇 and 𝑗=1, …, 𝑁_𝑄, she computes the blind squared-distance

        bd(𝑖,𝑗)=𝑒𝑑_𝑖𝑗+𝑡⁴_𝑖,𝑗 and the blind orientation difference bo(𝑖,𝑗)=𝑒_𝑖−𝜃^′_𝑗

        between 𝑀^′_𝑗∈𝑀^𝑄 and each 𝑀_𝑖∈𝑀^𝑇.