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Outsourced Computation Verification

Outsourced Computation Verification. Roy Luo, UC Berkeley Under the mentorship of Graham Cormode, AT&T Labs DIMACS REU 2010. A Quick Review. Large amounts of data, limited memory E.g. Router observing network traffic Want to answer questions about data going through

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Outsourced Computation Verification

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  1. Outsourced Computation Verification Roy Luo, UC Berkeley Under the mentorship of Graham Cormode, AT&T Labs DIMACS REU 2010

  2. A Quick Review • Large amounts of data, limited memory • E.g. Router observing network traffic • Want to answer questions about data going through • You(verifier) and a 3rd party (helper) • Both read entire data stream once, helper gives you answer and (interactive) proof of correctness • Proof must be verifiable in limited space

  3. The Frequency Moment Problem • Data comes as updates to vector [a1…au] • Want to calculate , k>0 • Can generalize solution of this problem to solve inner product, range-sum

  4. One Round Solution Interpolating polynomial f(x,y) defined over finite field Zp V calculates f(1,r)…f(v,r) online H sends s(x) = j[v] f(x, j)k V verifies s(r)=f(1,r)k+…+f(v,r)k If s(x) checks out, return i  [h] s(i)

  5. Resource Complexity (1-round) Let length of vector = n = h•v Verifier Space: O(v•Log(p)), v=O(√n) Helper Space: O(n) Communication: O(k•h•Log(p)) for the kth frequency moment, h=O(√n)

  6. 1-Round Time Complexity Quadratic in length of vector

  7. Multi-Round Protocol Index vector using {0,1}d in d = Log(N) dimensional space Interpolate with d-variate polynomial f(x1 … xd) in Zp Verifier picks [r1 … rd]  [p]d, and calculates fk(r1, r2, … rd) online

  8. Multi-Round Protocol, cont. • Round 1: • H sends g1(x1)=x2…xdfk(x1, x2…xd), • V sends r1 • Round i: • H sends gi(xi) = xi+1…xdfk(r1, r2…ri-1, xi, xi+1…xd) • V checks gi-1(ri-1) = gi(0) + gi(1), sends ri • Round d: • H sends gd(xd) = fk(r1, … rd-1, xd) • V checks gd(rd) = fk(r1, r2, … rd) • Dishonest H can only fool V with prob. <O(Log(n)/p)

  9. Verifier’s Space Complexity V must remember [r1 … rd] and f(r1 … rd)

  10. Prover’s Space Complexity P must remember entire vector

  11. Communication Complexity V sends[r1 … rd], P sends d=Log(n) polynomials of degree k

  12. Time Complexity (naïve)

  13. Time Complexity (better) O(nLog(n))

  14. Directions for Further Research Can existing protocols be extended to other problems or generalized? What other problems require protocols? What class of problems can be solved in this model?

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