Commitment Schemes

Commitment schemes are a way for one counterparty to commit to a value such that the value committed remains private, but can be revealed at a later time when the committing party divulges a necessary parameter of the commitment process. Strong commitment schemes must be both information hiding and computationally binding.

The Pedersen commitment sheme allows a sender to create a commitment to a secret value. They may then later open the commitment and reveal the value in a verifiable manner that binds them to their commitment. A commitment shceme consists of a three stages:

1. Setup
2. Commit
3. Open

example :: IO Bool
example = do
  -- Setup commitment parameters
  (a, cp) <- setup 256

  -- Commit to the message using paramaters: Com(msg, cp)
  let msg = 0xCAFEBEEF
  Pedersen c r <- commit msg cp

  -- Open and verify commitment: Open(cp,c,r)
  pure (open cp c r)

Pedersen commitment scheme has the following properties:

1. Hiding: A dishonest party cannot discover the honest party's value.
2. Binding: A dishonest party cannot open his or her commitment in more than one way
3. Non-correlation: A dishonest party cannot commit to a value that is in some significant way correlated to the honest party's value.

Using Pedersen commitments we implement mutually independent commitments system, a secure multiparty communication protocol in which counterparties can commit to arbitrary messages or data in a binding way.

Pedersen commitments are also additionally homomorphic, such that for messages m0 and m1 and blinding factors r0 and r1 we have:

Commit(m0; r0) * Commit(m1; r1) = Commit(m0 + m1; r0 + r1)

Pedersen Commitments (Elliptic Curves)

A more efficient implementation of the Pedersen Commitment scheme arises from Elliptic Curve Cryptography (ECC) which is based on the algebraic structure of elliptic curves over finite (prime) fields. Using ECC, the commitment scheme computations require fewer bits and as a result yields a much faster commitment phase.

Given a secure elliptic curve (e.g. secp256k1), a Pedersen commitment can be implemented using the same interface as usual but instead of prime field modular exponentiation, EC point multiplication and addition are used. The use of EC Pedersen commitments is almost exactly the same as the general prime field implementation:

example :: IO Bool
example = do
  -- Setup commitment parameters
  (a, cp) <- ecSetup Nothing -- SECP256k1 is used by default

  -- Commit to the message using paramaters: Com(msg, cp)
  let msg = 0xCAFEBEEF
  ECPedersen c r <- ecCommit msg cp

  -- Open and verify commitment: Open(cp,c,r)
  pure (ecOpen cp c r)

Additionally, the EC Pedersen Commitment implementation is also additively homomorphic in two ways:

Commit(x, r1) + Commit(y, r2) = Commit(x + y, r1 + r2)

and given a scalar n:

Commit(x,r) + n = Commit(x + n,r)

Vector Pedersen Commitments (Elliptic Curves)

The Vector Pedersen Commitment is a more powerful variant of the previous Pedersen commitment. It commits to a vector v instead of a scalar. This extended form is defined as:

C' = rH + (v₁G₁ + v₂G₂ + ... + v_nG_n)

where v₁, v₂, ..., v_n are scalars that multiply each point G₁, G₂, ..., G_n respectively in the elliptic curve. It is the result of the dot product between two vectors v and G of arbitrary large number of elements. Each element G_i is a NUMS ("Nothing Up My Sleeve") generator that can be created using a hash function H such that H(encode(G) || i) and a "coerce-hash-to-point" function to construct the point from the randomized hash value.

The new commitment C' is still a point in the curve and a valid Pedersen commitment. It also holds the hiding and binding properties and the same additive homomorphic properties as the Pedersen Commitment:

Commit(v, r1) + Commit(w, r2) = Commit(w + v, r1 + r2)

Commit(v, r) + w = Commit(v + w, r)

where v and w are now vectors.

References:

1. Pedersen, Torben Pryds. "Non-interactive and information-theoretic secure verifiable secret sharing." Annual International Cryptology Conference. Springer Berlin Heidelberg, 1991. APA
2. Liskov, Moses, et al. "Mutually independent commitments." International Conference on the Theory and Application of Cryptology and Information Security. Springer Berlin Heidelberg, 2001. APA
3. Blum, Manuel, and Silvio Micali. "How to generate cryptographically strong sequences of pseudorandom bits." SIAM journal on Computing 13.4 (1984): 850-864.

Usage

$ stack build
$ stack repl
> :load example/Example.hs

License

Copyright 2017-2018 Adjoint Inc

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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    http://www.apache.org/licenses/LICENSE-2.0

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