# bLSAG ring signatures overview
*2022-07-20*
> Note: I’m not a mathematician, I’m just an amateur on math. These notes are just an attempt to try to sort the notes that I took while learning abut bLSAG.
bLSAG: Back's Linkable Spontaneous Anonymous Group signatures
- signer ambiguity
- linkability
- unforgeability
### Setup
Let $G$ be the generator of an EC group.
We use a hash function $\mathcal{H}_p$, which maps to curve points in EC, and a normal hash $\mathcal{H}_n$, which maps to $\mathbb{Z}_p$.
Signer's key pair: $k_{\pi}$, s.t. $K_{\pi} = k_{\pi} \cdot G \in \mathcal{R}$, with secret index $\pi$.
Set of Public Keys: $\mathcal{R} = \{ K_1, K_2, \ldots, K_n \}$
```python
def new_key():
k = F.random_element()
K = g * k # g is the generator of the EC group
return K
```
### Signature
1. compute key image: $\tilde{K} = k_{\pi} \mathcal{H_p} ( K_{\pi}) \in G$
```python
key_image = k * hashToPoint(K)
```
2. Generate $\alpha \in^R \mathbb{Z}_p$, and $r_i \in^R \mathbb{Z}_p$, for $i \in \{1, 2, \ldots, n \}$, with $i \neq \pi$
- $r_i$ is used for the fake responses
```python
a = F.random_element()
r = [None] * len(R)
for i in range(0, len(R)):
if i==pi:
continue
r[i] = mod(F.random_element(), p)
```
3. Compute $c_{\pi + 1} = \mathcal{H}_n ( m, [\alpha G], [\alpha \mathcal{H}_p(K_{\pi})])$
```python
c[pi1] = hash(R, m, a * g, hashToPoint(R[pi]) * a, p)
```
4. for $i=\pi + 1, \pi +2, \ldots, n, 1, 2, \ldots, \pi -1$, calculate, replacing $n+1 \rightarrow 1$
$$
c_{i+1} = \mathcal{H}_n (m, [r_i G + c_i K_i], [r_i \mathcal{H}_p (K_i) + c_i \tilde{K}])
$$
- Notice that (from step 3 & 4):
$\alpha \mathcal{H}_p (K_{\pi}) = r_{\pi} \mathcal{H}_p (K_{\pi}) + c_{\pi} \cdot (\tilde{K})$,
where $\tilde{K}= k_{\pi} \mathcal{H_p} ( K_{\pi})$, so:
$\alpha \mathcal{H}_p (K_{\pi}) = r_{\pi} \mathcal{H}_p (K_{\pi}) + c_{\pi} \cdot (k_{\pi} \mathcal{H}_p(K_{\pi}))$
which is equal to,
$\alpha \cdot \mathcal{H}_p (K_{\pi}) = (r_{\pi} + c_{\pi} \cdot k_{\pi}) \cdot \mathcal{H}_p(K_{\pi})$
From where we can see: $\alpha = r_{\pi} + c_{\pi} \cdot k_{\pi}$
which we can rearrange to
$r_{\pi} = \alpha - c_{\pi} \cdot k_{\pi}$.
```python
for j in range(0, len(R)-1):
i = mod(pi1+j, len(R))
i1 = mod(pi1+j +1, len(R))
c[i1] = hash(R, m, r[i] * g + c[i] * R[i],
r[i] * hashToPoint(R[i]) + c[i] * key_image, p)
```
6. Define $r_{\pi} = \alpha - c_{\pi} k_{\pi} \mod{p}$
```python
r[pi] = mod(a - c[pi] * k, p)
```
Signature: $\sigma(m) = (c_1, r_1, \ldots, r_n)$, with key image $\tilde{K}$ and ring $\mathcal{R}$.
- $len(\sigma(m)) = 1+n$
```python
return [c[0], r]
```
#### Step by step (simplified):
- Generate $r_i \in^R \mathbb{Z_p}$
- Compute $c_{i+1}$ from $r_i$
- Link $r_{\pi}$ with $c_{\pi}$
*You can scroll down the images through the step-by-step diagrams.*
It reminds in some way to the approach to close a box like the one in the picture:
### Verification
1. check $p \tilde{K} \stackrel{?}{=} 0$
- to ensure that $\tilde{K} \in G$ (and not in a cofactor group of $G$)
2. for $i = 1, 2, \ldots, n$, replacing $n+1 \rightarrow 1$
$$
c'_{i+1} = \mathcal{H}_n (m, [r_i G + c_i K_i], [r_i \mathcal{H}_p (K_i) + c_i \tilde{K}])
$$
3. check $c_1 \stackrel{?}{=} c'_i$
```python
c[0] = c1
for j in range(0, len(R)):
i = mod(j, len(R))
i1 = mod(j+1, len(R))
c[i1] = hash(R, m, r[i] * g + c[i] * R[i],
r[i] * hashToPoint(R[i]) + c[i] * key_image, p)
assert c1 == c[0]
```
## Links
Toy implementation:
- Sage: https://github.com/arnaucube/math/blob/master/ring-signatures.sage
- Rust: https://github.com/arnaucube/ring-signatures-rs
Resources:
- *"Zero to Monero"* - https://web.getmonero.org/library/Zero-to-Monero-2-0-0.pdf
(section *"3.4 Back’s Linkable Spontaneous Anonymous Group (bLSAG) signatures"*)
