Oblivious transfer

In cryptography, an oblivious transfer (OT) protocol is a type of protocol in which a sender transfers one of potentially many pieces of information to a receiver, but remains oblivious as to what piece (if any) has been transferred. In cryptography, an oblivious transfer (OT) protocol is a type of protocol in which a sender transfers one of potentially many pieces of information to a receiver, but remains oblivious as to what piece (if any) has been transferred. The first form of oblivious transfer was introduced in 1981 by Michael O. Rabin.1 In this form, the sender sends a message to the receiver with probability 1/2, while the sender remains oblivious as to whether or not the receiver received the message. Rabin's oblivious transfer scheme is based on the RSA cryptosystem. A more useful form of oblivious transfer called 1–2 oblivious transfer or '1 out of 2 oblivious transfer', was developed later by Shimon Even, Oded Goldreich, and Abraham Lempel,2 in order to build protocols for secure multiparty computation. It is generalized to '1 out of n oblivious transfer' where the user gets exactly one database element without the server getting to know which element was queried, and without the user knowing anything about the other elements that were not retrieved. The latter notion of oblivious transfer is a strengthening of private information retrieval, in which the database is not kept private. Claude Crépeau showed that Rabin's oblivious transfer is equivalent to 1–2 oblivious transfer.3 Further work has revealed oblivious transfer to be a fundamental and important problem in cryptography. It is considered one of the critical problems in the field, because of the importance of the applications that can be built based on it. In particular, it is complete for secure multiparty computation: that is, given an implementation of oblivious transfer it is possible to securely evaluate any polynomial time computable function without any additional primitive.4 In Rabin's oblivious transfer protocol, the sender generates an RSA public modulus N=pq where p and q are large prime numbers, and an exponent e relatively prime to λ(N) = (p − 1)(q − 1). The sender encrypts the message m as me mod N. If the receiver finds y is neither x nor −x modulo N, the receiver will be able to factor N and therefore decrypt me to recover m (see Rabin encryption for more details). However, if y is x or −x mod N, the receiver will have no information about m beyond the encryption of it. Since every quadratic residue modulo N has four square roots, the probability that the receiver learns m is 1/2. In a 1–2 oblivious transfer protocol, the sender has two messages m0 and m1, and the receiver has a bit b, and the receiver wishes to receive mb, without the sender learning b, while the sender wants to ensure that the receiver receives only one of the two messages.The protocol of Even, Goldreich, and Lempel (which the authors attribute partially to Silvio Micali), is general, but can be instantiated using RSA encryption as follows. A 1-out-of-n oblivious transfer protocol can be defined as a natural generalization of a 1-out-of-2 oblivious transfer protocol. Specifically, a sender has n messages, and the receiver has an index i, and the receiver wishes to receive the i-th among the sender's messages, without the sender learning i, while the sender wants to ensure that the receiver receive only one of the n messages. 1-out-of-n oblivious transfer is incomparable to private information retrieval (PIR).On the one hand, 1-out-of-n oblivious transfer imposes an additional privacy requirement for the database: namely, that the receiver learn at most one of the database entries. On the other hand, PIR requires communication sublinear in n, whereas 1-out-of-n oblivious transfer has no such requirement.