Computing Availability over Time (CAT)

According to the core idea inside the previous sub- section POCI, we can then define the cluster com- puting power and online availability of computa- tional power (Computing Availability over Time, CAT ), the real sense of allocating the whole net- work computational power and mining interests. Again back to the node θ, if it claims to have ob- tained N computational power ϕ, then according to the principle mentioned above, it will accept N signatures to verify POCI, which is obviously a bit tedious, especially when this N is above the or- der of a thousand, it will greatly increase the proof workload. Therefore, the following integration opti- mization is proposed to integrate N signature func- tions into a new signature function Signχ, which is generated by the regular integration operator τ , i.e:

Eχ(m) = Signχ(m) = τ(Sign1 (m)+

(Sign2 (m) + ... + (SignN (m)) (10)

Eχ(m)is the encrypted message after integration of N signatures, and other verifying nodes verify it by the integration verification function Dχ of N public keys, where:

Dχ(x) = τ−1(D1(m)+(D2(m)+ ... +(DN(m))(11)

Considering the regularity properties of the integra- tion operator τ :

τ −1 Di(τEi(x)) = Di(Ei(x))（12）

τ −1 Di(τEj(x)) = 0(i j)（13）

It is not difficult to find out that if the following are satisfied:

Nm = Dχ(Eχ(m)) = D1(E1(m))+

D2(E2(m)) + ... + DN(EN(m)) (14)

then it is proved that θ has gained the computational power of Nϕ . If the total computational power of the current network is C, then the ratio η of θ to the total network computational power is:

This is the computational power under POCI. The computational power of each node will be serial- ized into segments, and the computational power of a chip is counted as a unit segment. Nϕ is ab- stractly sliced into N segments and labeled with serial numbers, as shown in Figure 2 below. A total segment consists of 1, 2, ... , n nodes form a total segment λ, and the length of the segment obtained by each node depends on the computational power size. And by applying VRF (Verifiable Random Function) function, the total fragment is random sampled at time interval T to get the serial number κ, then the node corresponding to the fragment of the chip serial number searched is the burst block node:

κ = VRF(T,λ) ( 16)

κ → ϕ(κ) ϕ → θ(ϕ) ( 17)

This mechanism shows that the node fragment with large computational power occupies more space and has a higher probability η of being chosen by VRF, thus achieving POCI.

Figure 2: Random sampling process based on VRF for com- putational fragments composed of nodes

However, in order to incentivize nodes to maintain their computational power, nodes that obtain the right to burst blocks need to perform CAT verification, which repeats previous POCI challenge, to prove again that they are in a highly computational power state at any moment t. Sim- ilarly, the digital signature of the computational chip and the verification session of each node are performed again, and the block reward is obtained after completion. In this way, POCI computational proof and CAT complete the implementation of the POCI mechanism of the entire Utility network in a way of low-energy, low-cost and high-efficiency.