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 M_c, 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(F_q). The set of points E(F_q) 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 K_c 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, ..., M_n-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 ∈ Z_p 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 ∈ Z_n, samples b ∈ Z_p and replies R = cA + bG. Alice aborts if R is not a valid point in the curve.

Key derivation

1. For all e ∈ Z_n, Alice computes k_e = aR - eT. She now has a vector of keys (k₀, ..., k_n-1).

2. Bob computes k_R = bA.

We can see that the key k_e = aR - eT = abG + (c - e)T. If e = c, then k_c = abG = bA = k_R. Therefore, k_R = k_c 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 ∈ Z_p 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 ∈ Z_n, samples bⁱ ∈ Z_p and replies Rⁱ = cⁱA + bⁱG. Alice aborts if Rⁱ is not a valid point in the curve.

Key derivation

1. For all eⁱ ∈ Z_n, Alice computes k_eⁱ = aRⁱ - eⁱT. She now has a vector of vectors of keys (k₀ⁱ, ..., k_n-1ⁱ).

2. Bob computes k_Rⁱ = bⁱA.

We can see that the key k_eⁱ = aRⁱ - eⁱT = abⁱG + (cⁱ - eⁱ)T. If e = c, then k_cⁱ = abⁱG = bⁱA = k_Rⁱ. Therefore, k_Rⁱ = k_cⁱ 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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