Privacy-Preserving Restricted Boltzmann Machine

Algorithm 2

Securely computing the product of two integers [12].

Initialize

Step 1.

 Party 𝐴 first generates a random number 𝑅 and computes 𝑀⋅𝑖−𝑅 for each 𝑖, 𝑖 is the possible input of Party 𝐵.

 Then define 𝑚_𝑖=𝑀⋅𝑖−𝑅, 𝑚_𝑖 is the plain text. Then Party 𝐴 encrypts each 𝑚_𝑖

 using ElGamal scheme and gets 𝐸(𝑚_𝑖,𝑟_𝑖), where each 𝑟_𝑖 is a new random number.

 After that, Party 𝐴 sends each 𝐸(𝑚_𝑖,𝑟_𝑖) to Party 𝐵 in the increasing order of 𝑖.

Step 2.

 Party 𝐵 picks 𝐸(𝑚_𝑁,𝑟_𝑁), rerandomizes it, and sends 𝐸(𝑚_𝑁,𝑟^′)

 back to Party 𝐴, where 𝑟^′=𝑟_𝑁+𝑠, and 𝑠 is only known to Party 𝐵.

Step 3.

 Party 𝐴 partially decrypts 𝐸(𝑚_𝑁,𝑟^′) and sends the partially decrypted message to Party 𝐵.

Step 4.

 Party 𝐵 finally decrypts the message (by doing partial decryption on the already partially decrypted message)

  to get 𝑚_𝑁=𝑀⋅𝑁−𝑅.

Note that 𝑅 is only known to Party 𝐴 and 𝑚_𝑁 is only known to Party 𝐵. Furthermore, 𝑚_𝑁+𝑅=𝑀⋅𝑁.