add signature documentation page (#123)

Author: arnaucube

book/src/SUMMARY.md

@@ -10,6 +10,7 @@
 - [The backend structure of a POD]()
   - [Backend types](./backendtypes.md)
   - [MerkleTree](./merkletree.md)
+  - [Signature](./signature.md)
 - [Deductions](./deductions.md)
   - [Statements](./statements.md)
     - [Statements involving compound types and Merkle trees](./merklestatements.md)

book/src/signature.md

@@ -0,0 +1,41 @@
+# Signature
+
+
+Current signature scheme used is proof-based signatures using Plonky2 proofs, following [https://eprint.iacr.org/2024/1553](https://eprint.iacr.org/2024/1553) and [https://jdodinh.io/assets/files/m-thesis.pdf](https://jdodinh.io/assets/files/m-thesis.pdf). This comes from [Polygon Miden's RPO STARK-based](https://github.com/0xPolygonMiden/crypto/blob/d2a67396053fded90ec72690404c8c7728b98e4e/src/dsa/rpo_stark/signature/mod.rs#L129) signatures.
+
+In future iterations we may replace it by other signature schemes (either elliptic curve based scheme on a Golilocks-prime friendly curve, or a lattice based scheme).
+
+
+
+### generate_params()
+$pp$: plonky2 circuit prover params<br>
+$vp$: plonky2 circuit verifier params<br>
+return $(pp, vp)$
+
+### keygen()
+secret key: $sk \xleftarrow{R} \mathbb{F}^4$<br>
+public key: $pk := H(sk)$ [^1]<br>
+return $(sk, pk)$
+
+### sign(pp, sk, m)
+$pk := H(sk)$<br>
+$s := H(pk, m)$<br>
+$\pi = plonky2.Prove(pp, sk, pk, m, s)$<br>
+return $(sig:=\pi)$
+
+### verify(vp, sig, pk, m)
+$\pi = sig$<br>
+$s := H(pk, m)$<br>
+return $plonky2.Verify(vp, \pi, pk, m, s)$
+
+
+### Plonky2 circuit
+private inputs: $(sk)$<br>
+public inputs: $(pk, m, s)$<br>
+$pk \stackrel{!}{=} H(sk)$<br>
+$s \stackrel{!}{=} H(pk, m)$
+
+
+<br><br>
+
+[^1]: The [2024/1553 paper](https://eprint.iacr.org/2024/1553) uses $pk:=H(sk||0^4)$ to have as input (to the hash) 8 field elements, to be able to reuse the same instance of the RPO hash as the one they use later in the signature (where it hashes 8 field elements).

book/src/signedpod.md

@@ -5,7 +5,7 @@ A SignedPod consists of the following fields:
 	- it can only contain [`ValueOf` statements](./statements.md).
 	- the Signer's public key is one of the key-values in the `kvs`.
 - `id`: the Root of the `kvs` MerkleTree
-- `signature`: a signature over the `id`
+- `signature`: a [signature](./signature.md) over the `id`
 - `signer`: the public key attached to the digital signature `signature`
 - `type`: the constant `SIGNATURE`