bitcoinsecp256k1-schnorr

This Rust crate is a direct translation from C++ to Rust of the Schnorr signature implementation used in the Bitcoin system.

Notice

This crate is part of a direct translation from C++ to Rust of the Bitcoin Core. As such, some of the function bodies may still be in the process of translation. Please note that this system will become testable once the translation process is complete.

Mathematical Ideas

Schnorr signatures are a type of digital signature that are based on the discrete logarithm problem. The signature scheme was first proposed by Claus-Peter Schnorr in 1989, and has since become an important tool in cryptography, including its use in the Bitcoin system.

The Schnorr signature scheme is based on the security of the discrete logarithm problem in a finite field. The scheme works by having the signer generate a random secret key, compute a public key using that secret key, and then sign a message using a combination of the secret key and the message. The signature can then be verified by anyone who knows the public key.

The tokens in this crate implement various functions related to the Schnorr signature scheme, including the computation of the nonce used in the signing process, the signing and verification of messages, and various tests to ensure the correctness of the implementation.

Overall, the Schnorr signature scheme is an important component of the Bitcoin system, and understanding its implementation is crucial for anyone working with the system. The implementation of this crate in Rust provides a fast and efficient implementation of the Schnorr signature scheme, and can be used as a building block for other cryptographic applications.

22> thanks -- could you please describe the schnorr signature scheme in mathematical detail? what are its mathematical properties?

Sure! The Schnorr signature scheme is a digital signature scheme based on the difficulty of computing discrete logarithms in finite fields.

The scheme consists of three steps: key generation, signature generation, and signature verification.

Key Generation

In this step, the signer generates a public-private key pair. Specifically, the signer chooses a secret integer $x$ (the private key) uniformly at random from the interval $[1, n-1]$, where $n$ is the order of the base point $G$ of the elliptic curve being used. The signer then computes the corresponding public key $P = xG$, where $G$ is the base point.

The public key $P$ is then shared with others who wish to verify signatures from the signer.

Signature Generation

To sign a message $m$, the signer follows the following steps:

1. Choose a random integer $k$ uniformly at random from the interval $[1, n-1]$.

2. Compute the point $R = kG$.

3. Compute the value $e = H(m | R)$, where $H$ is a cryptographic hash function.

4. Compute the value $s = k + xe \mod n$.

The signature for the message $m$ is the pair $(R, s)$.

Signature Verification

To verify a signature $(R, s)$ for a message $m$, the verifier computes:

1. The value $e = H(m | R)$.
2. The point $P' = sG - eR$.
3. If $P' = P$, the signature is valid. Otherwise, the signature is invalid.

The security of the Schnorr signature scheme is based on the difficulty of computing discrete logarithms in the underlying elliptic curve. In particular, the security of the scheme relies on the difficulty of computing the private key $x$ given the public key $P$, and the difficulty of computing the value $k$ used in the signature generation step given the signature $(R, s)$ and the message $m$.

One of the notable properties of the Schnorr signature scheme is that it is provably secure in the random oracle model, under the assumption that the discrete logarithm problem is hard in the underlying elliptic curve. Additionally, the Schnorr signature scheme has several advantages over other signature schemes, including its simplicity and efficiency, and the ability to aggregate multiple signatures into a single signature, which can be useful in certain applications.

23> what is the random oracle model? how do we use it to aggregate multiple signatures into a single signature? is it correct to say that schnorr scheme encapsulates the elliptic curve mathematics we were discussing before?

The random oracle model is a theoretical model that is often used to analyze the security of cryptographic protocols. In this model, a random function is assumed to exist, called the "random oracle," which maps inputs to random outputs. This random oracle is used as a substitute for a cryptographic hash function, which is used in practice.

The random oracle model is used to analyze the security of the Schnorr signature scheme because it relies on a hash function to compute the value $e$ used in the signature generation and verification steps. In practice, a cryptographic hash function such as SHA-256 is used as the hash function, but in the random oracle model, a random oracle is used instead.

One of the advantages of the Schnorr signature scheme is that it can be used to aggregate multiple signatures into a single signature, which can be useful in certain applications. This is achieved through a technique called "batch verification," where multiple signatures are verified simultaneously using a single computation.

To perform batch verification, the verifier collects all of the signatures and messages to be verified, and computes a single value $e' = H(m_1 | R_1) + \cdots + H(m_k | R_k)$, where $k$ is the number of signatures, $m_i$ is the $i$-th message, and $R_i$ is the $i$-th point in the signature. The verifier then computes the value $P' = s_1 G - e' R_1 + \cdots + s_k G - e' R_k$, where $s_i$ is the $i$-th value in the signature. If $P' = P$, where $P$ is the public key of the signer, then all of the signatures are valid.

Yes, it is correct to say that the Schnorr signature scheme encapsulates the elliptic curve mathematics we were discussing before, as it is a digital signature scheme based on the discrete logarithm problem in an elliptic curve.

24> what else should we know about it? any caveats?

One caveat with the Schnorr signature scheme is that it is subject to the same attacks as other digital signature schemes based on the discrete logarithm problem. In particular, if the discrete logarithm problem is ever solved for the elliptic curve used in the Schnorr scheme, then the security of the scheme would be compromised.

Another potential caveat is that the Schnorr scheme is not yet widely adopted in practice, and thus may not be as well-vetted as other signature schemes such as the widely used ECDSA. However, it is gaining traction in the Bitcoin community due to its potential advantages over ECDSA, such as its ability to be used for signature aggregation and its resistance to certain types of attacks.