Advanced Mathematics and Numerical Modeling of IoT

Class A

participants

Rounds

controller

Pedersen bit

commitment

module

Parameter

sequence

generator

Verifiable

parameter

distribution

module

Okamoto

signature

module

Class B

participants

Parameter

distributor

Figure 1: The structure diagram of model.

Proof.Differential equations are established for the rounds
(푟 0 ,푟 1 ,...,푟∗) respectively, based on the four conditions
mentioned above

Pr푘(푟 0 ,푟)=Pr{푁 (푟 0 ,푟)=푘}

=

[휆 ( 푟− 푟 0 )]

푘

푘!

푒−휆(푟−푟^0 ) (푟,푘∈푍).

(13)

The mathematical expectation is

퐸[푁(푟)−푁(푟 0 )]=휆(푟−푟 0 ). (14)

So the expectations rounds of this model are휆,eachtimethe
model convergence time complexity is푂(휆).

(3) Parameter Distributor. A machine can analog the behavior
of distributor (maker) and can be a trusted server in the
distributed network.

(4) Pedersen Bit Commitment Module. Pedersen bit commit-
ment protocol [ 20 ]isasecurityprotocoltakenascommit-
ment to the bit stream information. In each time of signature,

the system generates coefficients of homogeneous constant

coefficients differential equations, and the coefficients of

algebraic curved퐹(푥)with order푛 2 −1,whichcorrespond

to the participants in set퐵.Afterstoringthecoefficientsin

the binary bits formation, we note them as form of푚푖(푖 ∈

푍∧푚푖∈ {0, 1}), in the form of bits stream. The parameter

distributor is also attached with the bit commitment model

to prevent it from attacks.

Theorem 4.The model can detect whether the parameter

distributor is under attack or not.

Proof.The model adapts the Pedersen’s bit stream commit-

ment protocol.

Parameter distributor selects a random number

휌∈푅퐺퐹(푝max{푛^1 ,푛^2 })∗, timestamp information푡,andsecure

hash function퐻(푚푖,푡) (푖∈푍∧푚푖∈ {0, 1}).

To make bit stream and timestamp above hash process.

The primitive element of group퐺퐹(푝max{푛^1 ,푛^2 })is푔;publish

훿=푔휌푦퐻(푚푖,푡)mod푝max{푛^1 ,푛^2 } (푖 ∈ 푍 ∧ 푚푖∈{0, 1}).(15)

The triple(휌, 푚푖,푡)will be publish to the public, right after

the end of the signature process. Set퐴and set퐵participants