The GMR (Goldwasser, Micali and Rivest) one-time signature scheme uses a claw-free permutation [1]. It is one-time only method and once Alice releases her public key, she must show the private key, and which will show the mapping between the two. With a claw-free pair, we have a claw whhere \(g_0(x)=g_1(y)=z\), and where \(g_0\) and \(g_1\) are functions of \(x\) and \(y\), respectively. For example if we have \(g_0(x)=4x^2+3\) and \(g_1(y)=y+4\), the solution for this is \(x=1\) and \(y=3\). The claw would then be (\(1,3,7\)). For a claw-free permutation, \(g_0^{-1}\) and \(g_1^{-1}\) becomes computationally unfeasible. In this case we will use the example in [2], and where we create two prime numbers (\(p\) and \(q\)) to \(n=pq\), and the functions of:
\(g_0(x)= \begin{cases} x^2 \pmod n,& \text{if } x^2 \pmod n \lt \frac{n}{2}\\ -x^2 \pmod n,& \text{if } x^2 \pmod n \gt \frac{n}{2} \end{cases} \)
\(g_1(x)= \begin{cases} 4x^2 \pmod n,& \text{if } 4x^2 \pmod n \lt \frac{n}{2}\\ -4x^2 \pmod n,& \text{if } 4x^2 \pmod n \gt \frac{n}{2} \end{cases} \)
In this case, \(p=7\) and \(q=11\).