TNO PET Lab - Zero-Knowledge Proofs (ZKP) - Templates

The TNO PET Lab consists of generic software components, procedures, and functionalities developed and maintained on a regular basis to facilitate and aid in the development of PET solutions. The lab is a cross-project initiative allowing us to integrate and reuse previously developed PET functionalities to boost the development of new protocols and solutions.

The package tno.zkp.templates is part of the TNO Python Toolbox.

The research activities that led to this implementation is made possible by

• The Alliance of Privacy Preserving Detection of Financial Crime, consisting of De Volksbank, TMNL, CWI, ABN AMRO, Rabobank, TNO.
• The confidential 6G project
• The Early Research Project of TNO "Next generation crypto"

Limitations in (end-)use: the content of this software package may solely be used for applications that comply with international export control laws.
This implementation of cryptographic software has not been audited. Use at your own risk.

Documentation

Documentation of the tno.mpc.encryption_schemes.shamir package can be found here.

Install

Easily install the tno.mpc.encryption_schemes.shamir package using pip:

$ python -m pip install tno.zkp.templates

If you wish to run the tests you can use:

$ python -m pip install 'tno.zkp.templates[tests]'

Usage

This library contains the templates with which we can create a zero-knowledge proof(ZKP). The templates are protocols, which should to be inherited by any of the classes to support ZKPs. An example has been added in this repository. The example is the modulus linear form. The modulus linear form is a separate module, which is a basic building block that can be used to create ZKPs.

The ZKP library is based on Thomas Attema's dissertation Compressed $\Sigma$-protocol Theory, which can be found here. Many concepts are taken from it, and there will be references throughout the code to the dissertation. In this README the crucial concepts from the dissertation needed to use this library will be explained in short. If anything is unclear, feel free to raise an issue at the code repository.

Preliminaries

Before explaining how to use the library several concepts need to be explained. The concepts are used throughout the interface and knowing them beforehand makes it easier to use the library.

A zero-knowledge proof of knowledge (referred to as ZKP in this README) can be used to show you possess information without revealing the information itself. This can be used in many different settings, but currently uses with a distributed ledger are common.

Homomorphism

A homomorphism is a fundamental building block of the ZKP. The homomorphism evaluates a function on a vector of input elements. A homomorphism is represented in the literature as $\psi_n$, where $n$ is the length of the input vector.

The homomorphism maps the input vector from an abelian group $\mathbb{G}^n$ to a single element of another group $\mathbb{H}$. In particular the original domain and target image groups can have different group operators. In the dissertation a mapping is made from an additive group $\mathbb{G}$ to a multiplicative group $\mathbb{H}$.

Sigma Protocol

A sigma protocol is a three-step process which creates a zero knowledge proof. The three-step process uses a homomorphism, an input vector and random elements from a group.

The three steps are shown in the sequence diagram below. Each step is explained in this section.

Figure 1. Two parties exchange information in which the Prover wants to convince the Verifier that he knows secret input $x$ without revealing $x$.

Initial information

Before starting a sigma protocol certain pieces of information are calculated. The information consists of the following:

• Private input: $x$ secret of the prover
• Public input: $P$ and the homomorphism $\psi \in \texttt{Hom}(\mathbb{G}^n,\mathbb{H})$
• Prover's claim: $P=\psi(x)$

First step

In the first a commitment is made by the prover. The commitment is made by generating a random input $r$. The random input $r$ is evaluated by the homomorphism $\psi$ giving $A$. $A$ is the commitment sent to the verifier.

Second step

The verifier sends a challenge $c$ to the prover. The challenge is a single element (as opposed to a vector). The challenge needs to be able to do the following operations:

• Multiply with homomorphism input, which is a vector of elements from group $\mathbb{G}$. This operation is needed to create the response mentioned in the third step.
• Take to the power of the resulting group $\mathbb{H}$. This operation is needed to verify the proof.

In this library the homomorphism maps input from additive group operation to a group with multiplicative operations. Depending on your use-case and the chosen implementation different operations might be needed.

Third step

Create the response by calculating a new input for the homomorphism. The input is based on $r$, the challenge $c$ and the secret input $x$. The response is calculated as follows: $z=r+cx$.

Verification

The verifier can now check the proof of knowledge by testing if $\psi(z)$ is equal to $A\cdot P^c$.

Making it non-interactive

To be able to make the proof of knowledge non-interactive we need to be able to replace the challenge of the verifier. Replacing the challenge can not be done by just picking one as the prover. To replace the challenge we use the Fiat-Shamir transformation. The transformation uses a hashing algorithm to create the challenge.

Making the proof non-interactive prevents communication overhead and enables the option for the prover to create a proof of knowledge without the verifier being present. The verifier can then check the proof of knowledge on its own when the necessary information has been retrieved.

Creating a Sigma Protocol

To support the creation of a sigma protocol the template classes have been created. The template classes can be split into two categories.

The first category are the classes needed to create a basic sigma protocol. The basic sigma protocol creates a proof of knowledge in a non-interactive way. The StandardSigmaProtocol object contains all the information needed for the verification and none of the private information. The object can therefore be shared with the verifier for verification.

To create a StandardSigmaProtocol you need some random input and a homomorphism. For this example we will use the ModulusLinearForm homomorphism included in this package. The ModulusLinearForm implements the Homomorphism object and is located in the namespace tno.zkp.modulus_linear_form.

In the code snippet below we create a ModulusLinearForm corresponding to the following formula $1\cdot x_1 + 2\cdot x_2 + 3 \cdot x_3$ with the modulus $13$. The secret input is a random input generated by the homomorphism.

Generating the proof of knowledge is relatively straight forward. You call the method generate_proof with the homomorphism, the secret input and the hash function. The class will handle the process as described in the steps above.

To verify the proof of knowledge you only need to call the verify function.

from tno.zkp.modulus_linear_form import ModulusLinearForm
from tno.zkp.templates import StandardSigmaProtocol

homomorphism = ModulusLinearForm([1, 2, 3], 13)
secret_input_x = homomorphism.random_input()

proof_of_knowledge = StandardSigmaProtocol.generate_proof(
  homomorphism, secret_input_x, "sha256"
)

assert proof_of_knowledge.verify()

Compressing a Sigma Protocol

To compress a proof of knowledge there are some requirements on the homomorphism and the input. The requirements are enforced using the CompressibleHomomorphism and the CompressibleHomomorphismInput abstract classes.

Compressing a proof of knowledge makes the verification of the protocol cheaper. The cost savings occur due to a compression mechanism. The compression mechanism is described in detail in the dissertation.

The ModulusLinearForm from the previous example satisfies the requirements. Therefore, we can use the previous proof of knowledge for compression.

To apply the compression we need to use a compression mechanism. The compression mechanism from the dissertation has been implemented in this template. To apply it you need to do the following:

from tno.zkp.templates import full_compression

# compress the proof of knowledge as much as possible
compressed_protocol = full_compression(proof_of_knowledge)
assert compressed_protocol.verify()

The function full_compression reduces the ZKP from length $n$ until it can not be compressed anymore, which is a length of 1. The function used for the compression is called compression and is available to the user as well. The compression function halves the length of the ZKP.