1. Practical Private Computation of Vector Addition-Based Functions Yitao Duan and John Canny Computer Science Division University of California, Berkeley PODC 2007, August 12, Portland OR

2. Overview A method for performing privacy preserving distributed computation of many algorithms that is practical and secure in a realistic threat model at large scale Provably strong (information-theoretic) privacy Efficient ZKP to deal with cheating users

3. Model A few collaborating data miners mining data from n users Each user has an m-dimensional vector Realistic scale: m, n large (10 3 – 10 6 ) Threat: data miners are passive, users are allowed to cheat arbitrarily Challenge: standard cryptographic tools not feasible at this scale

4. Our Results Private computation based on secret sharing using addition only steps Private addition is much simpler than multiplication The main computation is in small field (32 or 64 bits) – private computation has the same cost as regular arithmetic A lot of (nonlinear) algorithms can be done with addition: Regression, Classification, Bayes net, Link analysis, SVD, EM. An extremely efficient ZKP that the L2 norm of user vector is bounded by L (Only O(logm) large field operations)

5. An Efficient Proof of Honesty The server asks for N random projections of the user’s vector, the user proves the square sum of them is small. Projections are done in small field. The only large field operations are N encryptions and boundedness ZKP O(log m) public key crypto operations (instead of O(m)) to prove that the L-2 norm of an m-dim vector is smaller than L.

6. Acceptance/rejection probabilities (a) Linear and (b) log plots of probability of user input acceptance as a function of |d|/L for N = 50. (b) also includes probability of rejection. In each case, the steepest (jagged curve) is the single-value vector (case 3), the middle curve is Zipf vector (case 2) and the shallow curve is uniform vector (case 1)

7. Performance (a) Verifier and (b) prover times in seconds with N = 50, where (from top to bottom) L has 40, 20, or 10 bits. The x-axis is the vector length m.

8. More Info Code available for download, soon. duan@cs.berkeley.edu http://www.cs.berkeley.edu/~duan Thank you!