Roy Luo, UC BerkeleyUnder the mentorship of

Graham Cormode, AT&T Labs

DIMACS REU 2010

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

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

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)

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)

Quadratic in length of vector

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

Round 1:

• H sends g1(x1)=x2…xd fk(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)

V must remember [r1 … rd] and f(r1 … rd)

P must remember entire vector

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

O(nLog(n))

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?