# RSA Schnorr Signatures

Maurer’s 2009 paper provides a generalization of schnorr signatures to a large class of situations. Essentially, any time you have a group homomorphism $\varphi : G \to H$, you can provide a zero-knowledge protocol to prove:

$$\Pi(X ; x) := \varphi(x) = X$$

(with $x$ kept secret).

In the case of a cyclic group of prime order $q$, with generator $G$, and the following homomorphism:

\begin{aligned} &\varphi : \mathbb{F}_q \to \mathbb{G}\cr &\varphi(x) := x \cdot G \end{aligned}

We have the usual Schnorr sigma protocol, which leads to Schnorr signatures after a Fiat-Shamir transform.

# RSA

We can use the following group homomorphism, inspired by RSA:

\begin{aligned} &\varphi : \mathbb{Z}/(N)^* \to \mathbb{Z}/(N)^*\cr &\varphi(m) := m^e \end{aligned}

Here, $(N, e)$ is an RSA public key.

Now, this is evidently a group homomorphism, since:

$$(a \cdot b)^e = a^e \cdot b^e$$

Furthermore, this homomorphism is one way, provided we don’t know the factorization of $N$, and we think RSA is hard.

# The Signature Protocol

For the memes, let’s explicitly describe the signature scheme.

Key-Generation

Generate random primes $p, q$ of half the desired modulus size. Let $N = p \cdot q$, and pick $e$ such that $\text{gcd}(e, (p - 1)(q - 1)) = 1$.

Pick a random $x\xleftarrow{R} \mathbb{Z}/(N)^*$, and then set $X \gets s^e \mod N$.

$x$ is the private key.

$(N, e, X)$ is the public key.

Signing \begin{aligned} k &\xleftarrow{R} \mathbb{Z}/(N)^*\cr K &\gets k^e \mod N\cr c &\gets H(N, e, X, K, m)\cr r &\gets k \cdot x^c \mod N\cr (K&, r) \end{aligned}

Verification $$r^e \stackrel{?}{\equiv} K \cdot X^{H(N, e, X, K, m)}\mod N$$

Trust me :)