Oblivious Transfer (OT) is a cryptographic primitive in which a sender transfers some of potentially many pieces of information to a receiver. The sender doesn’t know which pieces of information have been transferred.

1-out-of-2 OT

Oblivious transfer is central to many of the constructions for secure multiparty computation. In its most basic form, the sender has two secret messages as inputs, m0 and m1; the receiver has a choice bit c as input. At the end of the 1-out-of-2 OT protocol, the receiver should only learn message Mc, while the sender should not learn the value of the receiver’s input c.

The protocol is defined for elliptic curves over finite fields E(Fq). The set of points E(Fq) is a finite abelian group. It works as follows:

1. Alice samples a random a and computes A = aG. Sends A to Bob
2. Bob has a choice c. He samples a random b.
   • If c is 0, then he computes B = bG.
   • If c is 1, then he computes B = A + bG.

Sends B to Alice

3. Alice derives two keys:
   • K0 = aB
   • K1 = a(B - A)

It’s easy to check that Bob can derive the key Kc corresponding to his choice bit, but cannot compute the other one.

1-out-of-N OT

The 1-out-of-N oblivious transfer protocol is a natural generalization of the 1-out-of-2 OT protocol, in which the sender has a vector of messages (M0, …, Mn-1). The receiver only has a choice c.

We implement a protocol for random OT, where the sender, Alice, outputs n random keys and the receiver, Bob, only learns one of them. It consists on three parts:

Setup

Alice samples a ∈ Zp and computes A = aG and T = aA, where G and p are the generator and the order of the curve, respectively. She sends A to Bob, who aborts if A is not a valid point in the curve.

Choose

Bob takes his choice c ∈ Zn, samples b ∈ Zp and replies R = cA + bG. Alice aborts if R is not a valid point in the curve.

Key derivation

1. For all e ∈ Zn, Alice computes ke = aR - eT. She now has a vector of keys (k0, …, kn-1).

2. Bob computes kR = bA.

We can see that the key ke = aR - eT = abG + (c - e)T. If e = c, then kc = abG = bA = kR. Therefore, kR = kc if both parties are honest.

testOT :: ECC.Curve -> Integer -> IO Bool
testOT curve n = do

  -- Alice sets up the procotol
  (sPrivKey, sPubKey, t) <- OT.setup curve

  -- Bob picks a choice bit 'c'
  (rPrivKey, response, c) <- OT.choose curve n sPubKey

  -- Alice computes a set of n keys
  let senderKeys = OT.deriveSenderKeys curve n sPrivKey response t

  -- Bob only gets to know one out of n keys. Alice doesn't know which one
  let receiverKey = OT.deriveReceiverKey curve rPrivKey sPubKey

  pure $ receiverKey == (senderKeys !! fromInteger c)

k-out-of-N OT

1-out-of-N oblivious transfer can be generalised one step further into k-out-of-N. This is very similar in structure to the methods above comprising the same 3 parts:

Setup As above, Alice samples a ∈ Zp and computes A = aG and T = aA, where G and p are the generator and the order of the curve, respectively. She sends A to Bob, who aborts if A is not a valid point in the curve.

Choose Bob takes his choices ci ∈ Zn, samples bi ∈ Zp and replies Ri = ciA + biG. Alice aborts if Ri is not a valid point in the curve.

Key derivation

1. For all ei ∈ Zn, Alice computes kei = aRi - eiT. She now has a vector of vectors of keys (k0i, …, kn-1i).

2. Bob computes kRi = biA.

We can see that the key kei = aRi - eiT = abiG + (ci - ei)T. If e = c, then kci = abiG = biA = kRi. Therefore, kRi = kci if both parties are honest.

References:

1. Chou, T. and Orlandi, C. “The Simplest Protocol for Oblivious Transfer” Technische Universiteit Eindhoven and Aarhus University

Notation:

k: Lower-case letters are scalars. P: Upper-case letters are points in an elliptic curve. kP: Multiplication of a point P with a scalar k over an elliptic curve defined over a finite field modulo a prime number.

License

Copyright 2018 Adjoint Inc

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