Background

In B2 Sidechain, all transactions are being signed by the ECDSA signature algorithm which is described in following subsection. The raw transaction is first digested by the hash function (Keccak), and then the hash value is signed by the sender’s private key through ECDSA. The current version of Parlia consensus does not provide fast finality because one validator produces a block, and to make sure of the correctness of these operations, one has to wait for the long confirmation time, usually it is 2/3∗N+12/3*N+12/3∗N+1
  where NNN
 enotes the active validators. Aggregated signature mechanism with Parlia’s fast-finality can solve this problem because one can collect and convert many signatures into one aggregated signature and send only this aggregated signature to the chain. For the aggregated signature, some special elliptic curves such as BLS12381 or BN256 will be used.
Cryptographic Hash functions
A cryptographic hash function H:{0,1}∗→{0,1}kH: \{0,1\}^* \rightarrow \{0,1\}^kH:{0,1}∗→{0,1}k
 takes an arbitrary-length message and outputs a fixed-length output. A hash function has the following basic properties:
 
· Deterministic: Given mmm
, we always have x=H(m)x=H(m)x=H(m)
(the same input mmm
 always results in the same output xxx
).
· Efficient: it is very fast to compute the hash value for any given message.
· Pre-image resistance (one-wayness): For essentially all pre-specified outputs, it is computationally infeasible to find any input which hashed to that output.
· Second pre-image resistance It is computationally infeasible to find a second message that produces the same hash value.
· Collision resistant: It is also hard to find two arbitrary inputs xxx
 and yyy
 that hash to the same value, i.e., H(x)=H(y)H(x)=H(y)H(x)=H(y)
.
Digital Signatures: ECDSA Signing Algorithm
Let’s assume that nnn
 is order point, PPP
 and QQQ
 are two points on an elliptic curve, and GGG
 is a base point. The ECDSA signature algorithm can be described as follows:
Key generation: 
1. Select a random number ddd
 in the interval [1,n−1][ 1,n-1][1,n−1]
.
2. Compute Q=dGQ=dGQ=dG
 
3. Public key is QQQ
, private key is ddd
.
Signature generation: 
1. Select a random integer kkk
, 1≤k≤n1≤k≤n1≤k≤n
.
2. Compute  kG=(x1,y1)kG = (x_1,y_1)kG=(x1​,y1​)
 and convert x1x_1x1​
  to an integer  x^1\widehat{x}_1x1​
.
3. Compute r=x1 mod nr=x_1\ mod\ nr=x1​ mod n
. If r=0r=0r=0
 then go to step 1.
4. Compute k−1 mod nk^{-1}\ mod\ nk−1 mod n
.
5. Compute Hash(m)Hash(m)Hash(m)
 and convert this bit string to an integer eee
.
6. Compute s=k−1(e+dr) mod ns=k^{-1} (e + dr)\ mod\ ns=k−1(e+dr) mod n
. If s=0s=0s=0
 then go to step 1.
7. Signature for the message mmm
 is (r,s)(r,s)(r,s)
.
Signature verification: 
1. Verify that rrr
 and sss
 are integers in the interval [1,n−1][1,n-1][1,n−1]
.
2. Compute Hash(m)Hash(m)Hash(m)
 and convert this bit string to an integer eee
.
3. Compute w=s−1 mod nw = s^{-1}\ mod\ nw=s−1 mod n
.
4. Compute u1=ew mod nu_1 = ew\ mod\ nu1​=ew mod n
 and u2=rw mod nu_2 = rw\ mod\ nu2​=rw mod n
.
5. Compute X=u1G+u2QX = u_1G+u_2QX=u1​G+u2​Q
. 
6. If X=θX=\thetaX=θ
 then reject the signature. Otherwise, convert the xxx
-coordinate x1x_1x1​
 of XXX
 to an integer x^1\widehat{x}_1x1​
, and compute v=x^1 mod nv=\widehat{x}_1\ mod\ nv=x1​ mod n
.
Aggregated Signatures
BLS 12381
BLS (Boneh, Lynn, Shacham) is another digital signature introduced in 2001 and has an aggregated structure. Let e:G1×G2→G3e: \mathbb{G}_1 \times  \mathbb{G}_2 \rightarrow \mathbb{G}_3e:G1​×G2​→G3​
 be a pairing where G1, G2\mathbb{G_1},\ \mathbb{G_2}G1​, G2​
 are additive groups and G3\mathbb{G_3}G3​
 is a multiplicative group. Also, let G1,G2G_1, G_2G1​,G2​
 and G3G_3G3​
 are base elements of G1, G2\mathbb{G_1},\ \mathbb{G_2}G1​, G2​
 and G3\mathbb{G_3}G3​
 respectively.
 
Public and Private Key Pair (pk,sk)(pk,sk)(pk,sk)
:
· The private key sksksk
 to be used for signing is just a randomly chosen number between [1,r−1][1,r-1][1,r−1]
.
· The corresponding public key is pk=[sk]G1pk=[sk]G_1pk=[sk]G1​
.
Signing:
· To sign a message mmm
 we first need to map mmm
 onto a point in group G2\mathbb{G_2}G2​
. Let’s assume this mapping results in a G2\mathbb{G}_2G2​
 point H(m)H(m)H(m)
.
· We sign the message by calculating the signature σ=skH(m)\sigma=skH(m)σ=skH(m)
.
Verification:
Given a message mmm
, a signature σ\sigmaσ
, and a public key pkpkpk
, we want to verify that it was signed with the sksksk
.
· The signature is valid if, and only if, e(G1,σ)=e(pk,H(m))e(G_1,\sigma) = e(pk,H(m))e(G1​,σ)=e(pk,H(m))
.
Aggregation
One of the most important properties of BLS signatures is that they can be aggregated​
· To aggregate signatures, we just must add up the G2\mathbb{G}_2G2​
 points they correspond to: σaggregated=σ1+σ2+...+σn\sigma_{aggregated} = \sigma_1 + \sigma_2 + ...+ \sigma_nσaggregated​=σ1​+σ2​+...+σn​
.
· We also aggregate the corresponding G1G_1G1​
 public key point
 pkaggregated=pk1+pk2+...+pknpk_{aggregated}=pk_1+pk_2+...+pk_npkaggregated​=pk1​+pk2​+...+pkn​
.
· Verify that e(G1,σaggregated)=e(pkaggregated,H(m))e(G_1,\sigma_{aggregated})=e(pk_{aggregated}, H(m))e(G1​,σaggregated​)=e(pkaggregated​,H(m))
 to verify all the signatures together with just two pairings.
BN256 Curves
BN256 is basically the size of the prime number of the underlying field in G1, G2\mathbb{G}_1,\ \mathbb{G}_2G1​, G2​
 and G3\mathbb{G}_3G3​
. In a BN256 curve, G2\mathbb{G}_2G2​
 is basicallyE(GF(p)), G2E(GF(p)),\ \mathbb{G}_2E(GF(p)), G2​
 is a subgroup of E(GF(p12))E(GF(p^{12}))E(GF(p12))
 and  G3\mathbb{G}_3G3​
 is a subgroup of GF(p12)GF(p^{12})GF(p12)
. Elements of G1\mathbb{G}_1G1​
 requires the same number of bits as ppp
 for each elliptic curve point. We would like to highlight that not all prime-friendly curves support cofactor 1. This means that we may need a larger prime for a particular group order in some cases. Elements of G2\mathbb{G}_2G2​
 require the same as pkpkpk
 for each elliptic curve point coordinate, where kkk
 is the embedding degree of the curve. When using twisted curves, we can reduce this by 2, 3, 4, or 6 depending on the curve. BN curves have embedding degree 12 and support twists, therefore we can use elements with the same size as p126=p2p^{\frac{12}{6}} = p^2p612​=p2
.

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Contents

Cryptographic Hash functions

Digital Signatures: ECDSA Signing Algorithm

Aggregated Signatures

BLS 12381

BN256 Curves