# # Cryptography

This document provides an overview of the cryptographic building blocks that drand uses to generate publicly-verifiable, unbiased, and unpredictable randomness in a distributed manner. The drand beacon has two phases (a setup phase, and a beacon phase), which we describe below.

Generally, we assume that there are

Threshold cryptography has many applications, as it avoids single points of failure. One such application is cryptocurrency multi-sig wallets, where

**Note**: This document is intended for a general audience, and no in-depth cryptographic background knowledge is necessary to understand the presented concepts.

## # Setup phase

The purpose of the drand setup phase is to create a collective private, and public key pair shared among **collective public key**, together with a **private key share** of the **collective private key**. The key shares are computed such that no individual node knows the entire collective private key.

Each private key share can then be used to perform cryptographic threshold computations, such as generating threshold signatures, where at least

A DKG is performed in a fully distributed manner, avoiding any single points of failure. We give an overview of the different sub-components of the drand DKG implementation in the following subsections.

### # Secret sharing

Secret sharing is an important technique that many advanced threshold cryptography mechanisms rely on. Secret sharing allows one to split a secret value

Shamir's Secret Sharing (SSS) scheme is one of the most well-known and widely used secret sharing approaches, and it is a core component of drand. SSS works over an arbitrary finite field, but for simplicity, we use the integers modulo

**Share Distribution**: To share **dealer** first creates a polynomial

The dealer then creates one share

**Secret Reconstruction**: To recover the secret

Note that any subset of

### # Verifiable secret sharing

Shamir's Secret Sharing scheme assumes that the dealer is honest but this assumption might not always hold in practice.

A Verifiable Secret Sharing (VSS) scheme protects against malicious dealers by enabling participants to verify that their shares are consistent with those dealt to other nodes ensuring that the shared secret can be correctly reconstructed later on.

Drand uses Feldman's VSS scheme, an extension of SSS. Let

A cyclic group means there exists a generator

**Share Distribution**: In addition to distributing shares of the secret to the participants, the dealer also broadcasts commitments to the coefficients of the polynomial

These commitments enable each participant

**Secret Reconstruction**: The recovery of secret

### # Distributed key generation (DKG)

Although VSS schemes protect against a malicious dealer, the dealer still knows the secret itself. To create a collectively shared secret

Drand uses Pedersen's DKG scheme, which essentially runs

**Share Distribution**: Every participant

**Share Verification**: Each participant verifies the shares it receives as prescribed by Feldman's VSS scheme. If participant

**Share Finalization**: At the end of the protocol, the final share of participant

The collective public key associated with the valid shares can be computed as

**Note**: Even though the secret created using Pedersen's DKG can be biased, it is safe to use for threshold signing as shown by Rabin et al.

## # Beacon phase

In the previous section, we gave an overview of how to produce a collective distributed key pair via a DKG protocol. In this section, we describe how to use this collective key pair to generate publicly-verifiable, unbiased, and unpredictable randomness in a distributed manner

We first give an overview of pairing-based cryptography (PBC) which has become quite popular lately and is used in many modern consensus protocols or zero-knowledge proofs such as zk-SNARKs.

Afterward, we show how drand uses PBC in the randomness beacon generation phase for threshold Boneh-Lynn-Shacham (BLS) signatures.

Finally, we explain how drand links the generated threshold BLS signatures into a randomness chain.

### # Pairing-based cryptography

Pairing-based cryptography is based on *bilinear groups*

**Bilinearity**:

**Non-degeneracy**:

**Computability**: There exists an efficient algorithm to compute

Drand currently uses the BLS12-381 curve.

### # Randomness generation

To generate publicly-verifiable, unbiased, distributed randomness, drand utilizes threshold Boneh-Lynn-Shacham (BLS) signatures.

Below we first describe regular BLS signatures followed by the threshold variant.

#### # BLS signature

BLS signatures are short signatures that rely on bilinear pairings and consist only of a single element in

They are *deterministic* in the sense that a BLS signature depends only on the message and the signer's key unlike other signature schemes, such as
ECDSA, which requires a fresh random value for each signed message to be secure.

Put differently, any two BLS signatures on a given message produced with the same key are identical. In drand, we utilize this property to achieve unbiasability for the randomness generation. The BLS signature scheme consists of the following sub-procedures:

**Key Generation**: To generate a key pair, a signer first chooses a private key

**Signature Generation**: Let

To compute a BLS signature

**Signature Verification**: To verify that a BLS signature

It is easy to see that this equation holds for valid signatures since

#### # Signature threshold

The goal of a threshold signature scheme is to collectively compute a signature by combining individual partial signatures independently generated by the participants. A threshold BLS signature scheme has the following sub-procedures:

**Key Generation**: The

**Partial Signature Generation**: To sign a message *partial BLS signature*

**Partial Signature Verification**: To verify the correctness of a partial signature

**Signature Reconstruction**: To reconstruct the collective BLS signature

**Signature Verification**: To verify a collective BLS signature

Thanks to the properties of Lagrange interpolation, the value of

Additionally, Lagrange interpolation also guarantees that no set of less than

In summary, a threshold BLS signature

### # Chained randomness

The drand randomness beacon operates in discrete rounds

To extend this chain of randomness, each drand participant

Once at least

At that point, the random value of round

Afterward, drand nodes move to round

For round

In a nutshell, this construction of using the hash of a BLS signature can be considered as a Verifiable Random Function because of the uniqueness of the signature output combined with the usage of the random oracle (the hash function). When the input is unpredictable, the output of the random oracle is indistinguishable from a uniform distribution.

## # Conclusion

To summarize, drand is an efficient randomness beacon daemon that utilizes pairing-based cryptography,

To learn more about the background of drand, we refer to the corresponding slides.

Finally, for more formal background on public randomness, we refer to the research paper titled "Scalable Bias-Resistant Distributed Randomness" published at the 38th IEEE Symposium on Security and Privacy.

The threshold scheme described here is described and proven in this paper from Boldyreva.