# Sapling binding signature

(\oplus, \ominus) - jubjub curve points (diamonds in the Sapling spec, couldn’t find the right symbols)
(\boxplus, \boxminus) - scalar field operators
(+, -) - real world operators

• a + b \mod{n} = a \boxplus b

### Pedersen Commitment

• Com(v, rcv) = [v]V \oplus [rcv]R

Notice:

• Com(0, rcv) = [rcv]R
• Com(v, 0) = [v]V

### Setup

• v= \sum_i v_i^{old} - \sum_j v_j^{new} net value of spend transfers minus net value of output transfers
• We have n spend desc: cv_i^{old} =Com(v_i^{old}, rcv_i^{old})= [v_i^{old}]V \oplus [rcv_i^{new}]R
• And m output desc: cv_j^{new} =Com(v_j^{new}, rcv_j^{new})= [v_j^{new}]V \oplus [rcv_j^{new}]R

### Binding signature

An honest signer computes bsk as:
bsk = (\boxplus_i^n rcv_i^{old}) \boxminus (\boxplus_j^m rcv_j^{new})

And bvk is computed as
(\bigoplus_i^n cv_i^{old}) \ominus (\bigoplus_j^m cv_j^{new}) \ominus Com(v, 0) =(\bigoplus_i^n cv_i^{old}) \ominus (\bigoplus_j^m cv_j^{new}) \ominus ([v]V \oplus [0]R)=
(\bigoplus_i^n cv_i^{old}) \ominus (\bigoplus_j^m cv_j^{new}) \ominus [v]V

Now, let’s do some math magic with the expression for bvk here. Replace $cv$s with expressions:

1. (\bigoplus_i^n ([v_i^{old}]V \oplus [rcv_i^{old}]R)) \ominus (\bigoplus_j^m ([v_j^{new}]V \oplus [rcv_j^{new}]R)) \ominus [v]V=
Open the parentheses:
2. (\bigoplus_i^n [v_i^{old}]V \oplus \bigoplus_i^n[rcv_i^{old}]R) \ominus (\bigoplus_j^m [v_j^{new}]V \oplus \bigoplus_j^m[rcv_j^{new}]R) \ominus [v]V=
3. (\bigoplus_i^n [v_i^{old}]V) \oplus (\bigoplus_i^n[rcv_i^{old}]R) \ominus (\bigoplus_j^m [v_j^{new}]V) \ominus (\bigoplus_j^m[rcv_j^{new}]R) \ominus [v]V=
Now group by the generator (V or R):
4. ((\bigoplus_i^n [v_i^{old}]V) \ominus(\bigoplus_j^m [v_j^{new}]V)) \oplus
5. ((\bigoplus_i^n[rcv_i^{old}]R) \ominus (\bigoplus_j^m[rcv_j^{new}]R)) \ominus [v]V=
Now factor out the generators (here we change the operators because they aren’t curve points anymore, they are scalars):
6. [\boxplus_i^n v_i^{old}]V \ominus [\boxplus_j^m v_j^{new}]V) \oplus
[\boxplus_i^nrcv_i^{old}]R \ominus [\boxplus_j^m rcv_j^{new}]R \ominus [v]V=
And merge (keep switching to scalar operations):
7. [\boxplus_i^n v_i^{old} \boxminus \boxplus_j^m v_j^{new}]V) \oplus [\boxplus_i^n rcv_i^{old} \boxminus \boxplus_j^m rcv_j^{new}]R) \ominus [v]V=
Here we can see that one of the expressions corresponds to bsk:
8. [\boxplus_i^n v_i^{old} \boxminus \boxplus_j^m v_j^{new}]V) \oplus [bsk]R \ominus [v]V=
Rearrange and factor V out again:
9. [\boxplus_i^n v_i^{old} \boxminus \boxplus_j^m v_j^{new} \boxminus v]V) \oplus [bsk]R =
And if v \mod{r_J} = \boxplus_i^n [v_i^{old}] \boxminus \boxplus_j^m [v_j^{new}] (r_j – Jubjub scalar field):
10. [v]V \oplus [bsk]R \ominus [v]V=
11. [bsk]R = Com(0, bsk)

If the signature is valid with the validating key bvk, it proves the signer’s knowledge of bsk with relation to bvk: bvk = [bsk]R = [0]V \oplus [bsk]R = Com(0, bsk) (because when a signature is correct it makes us believe that the signer knows the secret key, the general idea of signatures).

### Wrong balance attack

Let v* = \sum_i v_i^{old} - \sum_j v_j^{new} - v

If v* \neq 0 \mod{r_J} (the balance field v in the tx doesn’t represent the actual balance change) then bvk = Com(v*, bsk):

• from the line 10 above: (\boxplus_i^n [v_i^{old}] \boxminus \boxplus_j^m [v_j^{new}] \boxminus v)V) \oplus [bsk]R = [v*]V \oplus [bsk]R = Com(v*, bsk)

If the adversary can find bsk' s.t. bvk = [bsk']R (has to be done to pass the signature check) then
bvk = Com(0, bsk') = Com(v*, bsk) which is impossible because of the binding property of the commitment scheme