Verifiable computation (VC) allows a computationally weak client to outsource the evaluation of a function on many inputs to a powerful but untrusted server. The client invests a large amount of off-line computation and gives an encoding of its function to the server. The server returns both an evaluation of the function on the client's input and a proof such that the client can verify the evaluation using substantially less effort than doing the evaluation on its own. We consider how to privately outsource computations using privacy preserving VC schemes whose executions reveal no information on the client's input or function to the server. We construct VC schemes with input privacy for univariate polynomial evaluation and matrix multiplication and then extend them such that the function privacy is also achieved. Our tool is the recently developed mutilinear maps. The proposed VC schemes can be used in outsourcing private information retrieval (PIR).

Abstract. A k-query locally decodable code (LDC) C : Σ n → Γ N encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even after a constant fraction of the coordinates have been corrupted. Yekhanin (2008) constructed a 3-query LDC of subexponential length, N = exp(exp(O(log n/ log log n))), under the assumption that there are infinitely many Mersenne primes. Efremenko (2009) constructed a 3-query LDC of length N 2 = exp(exp(O( √ log n log log n))) with no assumption, and a 2 r -query LDC of length N r = exp(exp(O( r log n(log log n) r−1 ))), for every integer r ≥ 2. Itoh and Suzuki (2010) gave a composition method in Efremenko's framework and constructed a 3 · 2 r−2 -query LDC of length N r , for every integer r ≥ 4, which improved the query complexity of Efremenko's LDC of the same length by a factor of 3/4. The main ingredient of Efremenko's construction is the Grolmusz construction for superpolynomial size set-systems with restricted intersections, over Z m , where m possesses a certain "good" algebraic property (related to the "algebraic niceness" property of Yekhanin (2008)). Efremenko constructed a 3-query LDC based on m = 511 and left as an open problem to find other numbers that offer the same property for LDC constructions. In this paper, we develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the "algebraic niceness" phenomenon in Z m . We show that every integer m = pq = 2 t − 1, where p, q and t are prime, possesses the same good algebraic property as m = 511 that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki's composition method. More precisely, we construct a 3 ⌈r/2⌉ -query LDC for every positive integer r < 104 and a (3/4) 51 · 2 rquery LDC for every integer r ≥ 104, both of length N r , improving the 2 r queries used by Efremenko (2009) and 3 · 2 r−2 queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.

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Catalano and Fiore propose a scheme to transform a linearly-homomorphic encryption into a homomorphic encryption scheme capable of evaluating quadratic computations on ciphertexts. Their scheme is based on the linearly-homomorphic encryption (such as Goldwasser-Micali, Paillier and ElGamal) and need to perform large integer operation on servers. Then, their scheme have numerous computations on the servers. At the same time, their scheme cannot verify the computations and cannot evaluate more than degree-4 computations. To solve these problems, we no longer use linearly-homomorphic encryption which based on number theory assumptions. We use label and pseudorandom function to encrypt message, which significantly reduce the computations on the servers and enable us to use homomorphic MACs technology to realize verifiable computations naturally. We also extend the method to construct d-server schemes, which allow the client to delegate degree-d computations on outsourced data.