Cryptanalysis of Morillo-Obrador polynomial delegation schemes

Verifiable computation (VC) allows a client to outsource (delegate) the computation of a function f on an input x to a server and then verify the server's results with substantially less time than computing f(x) from scratch. The security of VC requires no efficient adversary can persuade the client to accept any wrong results. Morillo and Obrador (PST 2013) proposed three VC schemes for outsourcing the computation of polynomial functions and claimed that all schemes are secure under the decisional subgroup membership assumption. The authors show a simple attack against the security of their first scheme and then extend the attack to the other two schemes. Morillo and Obrador (PST 2013) also claimed that their third scheme keeps the client's input private under the square root assumption. The authors show that this is not true under the standard definition of input privacy. In particular, a curious server can extract the client's input x, if the x is not too large. The authors' results show that Morillo-Obrador schemes cannot be used in the polynomial delegation.