Privacy-Preserving Restricted Boltzmann Machine

Algorithm 1

Securely computing the sigmoid function [12].

Initialize

Step 1.

 Party 𝐴 first generates a random number 𝑅 and computes 𝑦(𝑥₁+𝑖)−𝑅 for each 𝑖, 𝑖 is the possible inputs of Party 𝐵.

 We define 𝑚_𝑖=𝑦(𝑥₁+𝑖)−𝑅, 𝑚_𝑖 is the plain text. Party 𝐴 encrypts each 𝑚_𝑖 using the ElGamal scheme and gets 𝐸(𝑚_𝑖,𝑟_𝑖),

 where each 𝑟_𝑖 is a new random number. Party 𝐴 sends each 𝐸(𝑚_𝑖,𝑟_𝑖) in the increasing order of 𝑖.

Step 2.

 Party 𝐵 picks 𝐸(𝑚_𝑥2,𝑟_𝑥2), rerandomizes it and sends 𝐸(𝑚_𝑥2,𝑟^′) back to Party 𝐴,

 where 𝑟^′=𝑟_𝑥2+𝑠, and 𝑠 is only known to Party 𝐵.

Step 3.

 Party 𝐴 partially decrypts 𝐸(𝑚_𝑥2,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_𝑥2=𝑦(𝑥₁+𝑥₂)−𝑅. Note that 𝑅 is only known to Party 𝐴 and 𝑚_𝑥2 is only known to Party 𝐵.

 Furthermore, 𝑚_𝑥2+𝑅=𝑦(𝑥₁+𝑥₂)=𝑓(𝑥).