Pedersen-Commitments-in-EC-Form.md

Note: This chapter involves the ec_sum() method, which is only available in the upcoming Libbitcoin version 4 library (currently master branch).

Using EC operations, we can also generate Pedersen commitments in EC form, which can be expressed as the following:

C = r * H + a * G

The commitment C is binding, as neither the commited value a nor the random value r can be altered after the commitment has been made. It is also blinding, as the random value r ensures that no information about the commited value a can leak, even if two commitments are made to the same commited value a.

The committed value cannot be altered after commitment C is made, because H is a curve point with an unknown discrete log.

H = q * G
Where q is unknown:

If q were known, it could be factored out of H:

C = r * ( q * G ) + a * G = ( r * q + a ) * G

In this case, multiple possible values of r and a can be determined for a given commitment C. Therefore, r and a can be changed after the commitment has been made, rendering the commitment non-binding.

The following example demonstrates generating a Pedersen commitment in EC form in Libbitcoin. The full ready-to-compile code examples from this chapter can be found here.

// Example value for point h:
// h = q * G with unknown q.
auto point_h = base16_literal(
    "02b2138500d3754cd3009d8cc0bd5e7b89b0eb158594eef21ae7e4224bc1ff1a76");

// Verify point h is a valid EC point.
std::cout << verify(point_h) << std::endl;

// Generate Pedersen Commitment C;
// C = r * H + a * G

// Create random r.
data_chunk entropy_r(ec_secret_size);
pseudo_random_fill(entropy_r);
auto scalar_r = to_array<ec_secret_size>(entropy_r);

// r * H
ec_compressed left_point(point_h);
ec_multiply(left_point, scalar_r);

// a * G
ec_compressed right_point;
auto committed_a = base16_literal(
    "1aee6572a3590637cd3eaa95212aefb8c029b2d982feef2d38e53d0da2b5bae3");
secret_to_public(right_point, committed_a);

// C = r * H + a * G
point_list commitment_point_list = {left_point, right_point};
ec_compressed commitment_point;
ec_sum(commitment_point, commitment_point_list);

// Commitment point C.
std::cout << encode_base16(commitment_point) << std::endl;

Homomorphic Property of Pedersen Commitments

Homomorphism means that algebraic operations on commitments will also hold true for the binding and blinding factors they contain.

C( r1 , a1 ) + C( r2 , a2 ) = C( r1 + r2 , a1 + a2 )

This equality is demonstrated in the example below:

// Create a second commitment C(r2, a2).
// C2 = r2 * H + a2 * G
data_chunk entropy_r2(ec_secret_size);
pseudo_random_fill(entropy_r2);
auto scalar_r2 = to_array<ec_secret_size>(entropy_r2);
// r2 * H
ec_compressed left_point2(point_h);
ec_multiply(left_point2, scalar_r2);
// a2 * G
ec_compressed right_point2;
auto committed_a2 = base16_literal(
    "69f9e04fb736ab209fea2dcc97d70c8b0bfb778857517bee68a5eeda6d610a72");
secret_to_public(right_point2, committed_a2);
// C2 = r2 * H + a2 * G
point_list commitment_point_list2 = {left_point2, right_point2};
ec_compressed commitment_point2;
ec_sum(commitment_point2, commitment_point_list2);

// Create sum of two commitments.
// C from previous example: commitment_point.
// C + C2
point_list commitment_list = {commitment_point, commitment_point2};
ec_compressed commitment_sum;
ec_sum(commitment_sum, commitment_list);

// Now we generate a commitment from r+r2 and a+a2.
// C(r + r2, a + a2)
ec_secret scalar_sum_r(scalar_r);
ec_add(scalar_sum_r, scalar_r2);
ec_secret scalar_sum_a(committed_a);
ec_add(scalar_sum_a, committed_a2);

ec_compressed left_point_(point_h);
ec_multiply(left_point_, scalar_sum_r);

ec_compressed right_point_;
secret_to_public(right_point_, scalar_sum_a);

point_list commitment_list_ = {left_point_, right_point_};
ec_compressed commitment_sum_;
ec_sum(commitment_sum_, commitment_list_);

// Homomorphism holds.
// C + C2 = C(r + r2, a + a2)
std::cout << (commitment_sum == commitment_sum_) << std::endl;

The full ready-to-compile code examples from this chapter can be found here.