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Cicada

Cicada is a private on-chain voting protocol based on homomorphic time-lock puzzles.

Quickstart

Requires Foundry.

Install: forge install

Build: forge build

Differential tests: forge test --match-test testRef --ffi

Other tests: forge test --no-match-test testRef

How it works

At a high-level, our implementation adapts the linearly-homomorphic time-lock puzzle scheme described in (Malavolta and Thyagarajan, 2019), using exponential ElGamal instead of Paillier encryption.

Throughout the following descriptions, we take $\mathbb{Z}_N^* %$ to mean $\mathbb{Z}_N^* / {\pm 1}$, and we take $x \mod{N}$ to mean $\min\left(x \mod N, -x \mod N\right)$ –– see LibUint1024.normalize. We use $\mathbb{Z}_N^* / {\pm 1}$ to ensure the low-order assumption is not trivially false, as suggested in Section 6.

Homomorphic time-lock puzzles

A time-lock puzzle is a cryptographic puzzle that encapsulates some secret which can be recovered by performing $\mathcal{T}$ steps of (non-parallelizable) computation.

Time-lock puzzles are useful for voting schemes because they allow users to post their votes as a puzzle, ensuring it can eventually be revealed while keeping it secret during the election, a property called running-tally privacy. The goal is that users can cast votes without being influenced by other votes already cast. Time-lock puzzles are rather unique in the field of private voting schemes in that they achieve running-tally privacy without relying on tallying authorities, threshold encryption or any other trusted parties: anybody can solve a time-lock puzzle to ensure votes are revealed after the election.

A homomorphic time-lock puzzle is one which can be homomorphically manipulated without knowing the solution or backdoor. In particular, a linearly homomorphic time-lock puzzle allows one to add two puzzles, producing a new puzzle which encapsulates the sum of the the original two puzzles, or perform scalar multiplications on puzzles.

As the authors of the paper note, linearly homomorphic time-lock puzzles are particularly suitable primitive for private voting: ballots can be encoded as puzzles, and they can be homomorphically combined to obtain a puzzle encoding the final tally. This allows a single computation to recover the final tally, rather than solving a unique puzzle for every vote.

Exponential ElGamal

In the original HTLP paper, the linearly-homomorphic scheme is presented using Paillier encryption for the time-lock puzzle:

$$u := g^r \mod{N}$$

$$v := h^{r \cdot N} \cdot (1 + N)^s \mod{N^2}$$

The structure of the Paillier cryptosystem provides additive homomorphism and fast decoding of the secret $s$ once the $\mathcal{T}$ sequential squarings have been computed. However, ballot verification requires large exponentiations modulo $N^2$, which are prohibitively expensive (millions of gas) on most EVM chains.

Instead, we use exponential ElGamal, as follows:

$$u := g^r \mod{N}$$

$$v := h^{r} \cdot y^s \mod{N}$$

Exponential ElGamal provides additive homomorphism, but decoding the final tally requires brute-forcing the discrete log of $v \cdot h^{-r}$ base $y$. As such, it is only suitable if the expected final tally is reasonably small (e.g. $< 2^{32}$). Of course, this brute-force can be performed offline and the answer provided as a hint which is efficiently verified on-chain.

Public parameters

To instantiate an election, the following parameters are required:

• An RSA modulus $N$ (e.g. 1024-4096 bits). This could be a standard modulus, or one generated via multi-party computation. If someone knows the factorization of $N$, they would be able to quickly decrypt all ballots, undermining running-tally privacy. Future implementations could utilize class groups, removing the risk of this trapdoor. Our default implementation uses a 1024-bit modulus for efficiency. Note that, while this is well above the largest RSA modulus ever publicly factored (which was 829 bits) it is below the normally recommended size of 2048 bits for use with RSA encryption or signatures. However, we don't need long-term security in our application. Once an election is finished there is no risk if $N$ is factored in the future. Thus it is reasonable to use a relatively small modulus; this can be easily updated in the future if factoring algorithms improve.
• A "time" parameter $\mathcal{T}$, the number of sequential squarings required to reveal the contents of the time-lock puzzle. This parameter must be carefully chosen to ensure an adversary (potentially with hardware acceleration) can't decrypt votes during the ballot-casting period.
• Two randomly chosen generators $g$ and $y$ of $\mathbb{Z}_N^*$ with Jacobi symbol 1, and the precomputed value $h := g^{2^\mathcal{T}} \mod{N}$. It is possible to efficiently prove that $h$ was generated correctly using a Wesolowski proof or Pietrzak proof.

In the _createVote function, the time-lock puzzle associated with the vote's running tally is initialized to a value encoding 0. This could be done by setting both $u$ and $v$ to 1, but we instead initialize them to $g$ and $h$, respectively, so that all eight storage slots are populated (saving gas for the first ballot cast).

$$\text{tally}_\text{ct}.u := g^1 = g \mod{N}$$

$$\text{tally}_\text{ct}.v := h^1 \cdot y^0 = h \mod{N}$$

Together, $N, \mathcal{T}, g, h, y,$ and $y^{-1}$ comprise the public parameters of the vote, denoted pp in the smart contract. Note that $y^{-1}$ is included for efficiency (i.e. avoiding the need to compute a modular inverse on-chain).

Casting a ballot

The _createVote function takes the following parameters:

• string description (A human-readable description of the vote)
• uint64 startTime (When the voting period starts, as a Unix timestamp)
• uint64 votingPeriod (The length of the voting period, in seconds)

To cast a ballot, the voter provides a homomorphic time-lock puzzle encoding their choice (0 representing "no", 1 representing "yes"), and a zero-knowledge proof that the provided puzzle is a valid ballot.

Proving that the ballot is a valid is equivalent to the disjunction of two $\Sigma$-protocols of discrete-log equality, i.e. there exists some value $r$ such that:

$$(u = g^r \texttt{ AND } v = h^r) \texttt{ OR } (u = g^r \texttt{ AND } v \cdot y^{-1} = h^r)$$

where the former case represents a "no" ballot and the latter case represents a "yes" ballot. Protocol 1 describes the $\Sigma$-protocol for proving discrete-log equality, and Section 4 describes the standard construction for $\texttt{OR}$-composition of $\Sigma$-protocols.

A proof of ballot validity consists of eight values: $(a_0, b_0, t_0, c_0, a_1, b_1, t_1, c_1)$. Given a ballot puzzle $(u,v)$ and its proof of validity, the contract performs the $\Sigma$-protocol verification as follows:

$$c := \text{hash}(a_0, b_0, a_1, b_1, pp)$$

$$c_0 + c_1 \stackrel{?}{=} c \mod{2^{256}}$$

$$g^{t_0} \stackrel{?}{=} a_0 \cdot u^{c_0} \mod{N}$$

$$h^{t_0} \stackrel{?}{=} b_0 \cdot v^{c_0} \mod{N}$$

$$g^{t_1} \stackrel{?}{=} a_1 \cdot u^{c_1} \mod{N}$$

$$h^{t_1} \stackrel{?}{=} b_1 \cdot (v \cdot y^{-1})^{c_1} \mod{N}$$

Note that the protocol is made non-interactive using the Fiat-Shamir heuristic (the challenge $c$ is computed as the hash of the prover's first message).

If the proof of validity is successfully verified, the running tally is updated, leveraging the additive homomorphism of the time-lock puzzles:

$$\text{tally}_\text{ct}.u = \text{tally}_\text{ct}.u \cdot Z.u \mod{N}$$

$$\text{tally}_\text{ct}.v = \text{tally}_\text{ct}.v \cdot Z.v \mod{N}$$

Finalizing a vote

To finalize the result of a vote, someone needs to

1. perform the sequential squarings: $w := u^{2^T}$,
2. brute-force the discrete-log to recover the tally value: $\text{tally}_\text{pt} := \text{dlog}_y(v \cdot w^{-1})$, and
3. call the _finalizeVote function with the solution and associated proof of exponentiation.

For the proof of exponentiation, we will need a 256-bit prime. To make finalization non-interactive, we use the Fiat-Shamir heuristic. –– the prime must be a valid hash-to-prime output, as follows:

$$\text{hash}(\text{tally}_\text{ct}.u, w, pp, j)\mathbin{|}2^{255} \stackrel{?}{=} \ell$$

$$\text{Baillie-PSW}(\ell) \stackrel{?}{=} \text{true}$$

The primality test is instantiated using Baillie-PSW, but can be replaced with another probabilistic primality test (or e.g. Pocklington primality certificates).

The contract then verifies the Wesolowski proof of exponentiation as follows:

$$r := 2^\mathcal{T} \mod{\ell}$$

$$w \stackrel{?}{=} \pi^\ell \cdot u^r \mod{N}$$

Finally, the contract checks that the relation between $v, w,$ and the plaintext tally value (denoted $\text{tally}_\text{pt}$) holds:

$$v \stackrel{?}{=} w \cdot y^{\text{tally}_\text{pt}} \mod{N}$$

If all the checks pass, the vote is marked as finalized.

Anonymity via ZK set membership

The construction described above provides tally privacy –– the time-lock puzzle property keeps the tally private for the time paramter $\mathcal{T}$. However, each individual ballot is also a time-lock puzzle, encrypted under the same public parameters. This means that just as the tally can be decrypted (by sequential squaring and brute-forcing the secret), so can each individual ballot.

For some elections this may not be desirable; while we are satisfied with temporary tally privacy, we may want indefinite ballot privacy. To accomplish this, we can combine the homomorphic time-lock puzzle scheme with an anonymous voter eligibility protocol, instantiated by zero-knowledge set membership proofs. This way, even if a ballot is decrypted, all it would reveal is that someone voted that way –– which is already known from the tally.

We provide an example contract using Semaphore for anonymity, but you can plug in your ZK set membership solution of choice.

This repo

CicadaVote.sol

This contract contains the core logic of the homomorphic time-lock puzzle voting protocol. It is meant to be inherited and extended with application-specific logic, e.g. access control: who can create a vote, who can cast a ballot, etc.

The three main functions follow the intuitive lifecycle of a vote: _createVote, _castBallot, and _finalizeVote.

Semaphore extension

The SemaphoreCicada contract is an example of how one can extend CicadaVote with a ZK set membership protocol to achieve ballot privacy. Note that Semaphore could be replaced with some other ZK set membership module (e.g. Semacaulk) to the same effect.

Big integer arithmetic

The modular 1024-bit arithmetic needed for RSA group operations is implemented in LibUint1024.sol. We represent the numbers using uint256[4] types.

Most functions in the library can be generated for an arbitrary number size (uint256[N] for some N) using the LibUint.sol.jinja template and the corresponding script generate_lib_uint.py. The mulMod is a notable exception; the approach implemented LibUint1024.sol doesn't scale past uint256[4] due to stack size limitations.

LibPrime.sol

This library implements various primality tests: Miller-Rabin, Lucas, and Baillie-PSW. In addition, it implements Pocklington primality certificate verification, which guarantees (deterministically) that a number is prime.

The checkHashToPrime function (used in CicadaVote.sol) uses the Baillie-PSW primality test because it offers a good balance between gas efficiency and security.

Tests

LibUint1024.t.sol contains fuzz tests for the arithmetic functions in LibUint1024. These include differential tests, where the reference functions are written in Python (see big_math_reference.py).

The numbers used in LibPrime.t.sol were generated using https://bigprimes.org/.

VerifyBallot.t.sol and VerifySolution.t.sol contain tests for _verifyBallotValidity and _verifySolutionCorrectness, respectively. The tests were generated using generate_verify_ballot_test.py and generate_verify_solution_test.py. These scripts also give a sketch of how a user might generate these proofs in a production setting.

CicadaVote.t.sol contains an end-to-end test, and can be used to benchmark the gas cost of casting a ballot (using the --gas-report flag).

Disclaimer

These smart contracts are being provided as is. No guarantee, representation or warranty is being made, express or implied, as to the safety or correctness of the user interface or the smart contracts. They have not been audited and as such there can be no assurance they will work as intended, and users may experience delays, failures, errors, omissions or loss of transmitted information. THE SMART CONTRACTS CONTAINED HEREIN ARE FURNISHED AS IS, WHERE IS, WITH ALL FAULTS AND WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING ANY WARRANTY OF MERCHANTABILITY, NON- INFRINGEMENT OR FITNESS FOR ANY PARTICULAR PURPOSE. Further, use of any of these smart contracts may be restricted or prohibited under applicable law, including securities laws, and it is therefore strongly advised for you to contact a reputable attorney in any jurisdiction where these smart contracts may be accessible for any questions or concerns with respect thereto. Further, no information provided in this repo should be construed as investment advice or legal advice for any particular facts or circumstances, and is not meant to replace competent counsel. a16z is not liable for any use of the foregoing, and users should proceed with caution and use at their own risk. See a16z.com/disclosures for more info.