### Threshold signature for sr25519

The term sr25519 refers to the instatiation of Schnorr signature using the curve curve25519, the same curve as of EdDSA, which is specified in Schnorrkel. However, it additionally employs the method of point compression due to Ristretto to make makes Schnorr signatures over the Edward's curve more secure. Below we describe the curve parameter and its security and efficiency analysis.

The sr25519 scheme supports both forms of curve25519, i.e, its twisted Edward form and Montomery form. The twisted Edward form of curve25519 has been described in the previous Section. For the Montomery form, the curve can be written as $$E:by^2=x^3+ax^2+x$$ over $$\mathbb{F}_p$$ with order $$n$$, cofactor $$f$$ and base point $$G$$ are as follows:

• $$p=0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed$$

• $$a=0x76d06$$

• $$b=0x01$$

• $$n=0x1000000000000000000000000000000014def9dea2f79cd65812631a5cf5d3ed$$

• $$cofactor=1$$

• $$G=(0x09,0x20ae19a1b8a086b4e01edd2c7748d14c923d4d7e6d7c61b229e9c5a27eced3d9)$$

For security analysis, recall that the curve curve25519 achieves $$128$$ bit security level, as specified in the previous Section. However, since sr25519 additionally uses the point compression due to Ristretto, it is safe from the bug could lead to a double spend expoit of Monero

Finally, because sr25519 is actually the Schnorr signature instatiatied with the curve25519 and FROST is a threshold Schnorr signature scheme that can be instatiated by any curve where the discrete log problem is hard, we can instatiate the FROST threshold signature scheme of [KG20] with the parameters of sr25519 described in to achieve the MPC version of sr25519.