3.2. Improved Threshold Model.We adopt (푛 2 ,푡 2 ) threshold
structure constructed by matrix method.푡 2 players in set퐵
participate in the repeated games and recover the secret푆
using the published푛 2 −푡 2 points. As a result, the players
in set퐴can input푆after they get the general term formula
of homogeneous constant coefficient linear differential equa-
tion.
Make two field extensions:
[퐺퐹 (푝푛^1 ) : 퐺퐹 (푞)] = [퐺퐹 (푝푛^1 ):퐺퐹(푝)][퐺퐹(푝):퐺퐹(푞)],[퐺퐹 (푝푛^2 ) : 퐺퐹 (푞)] = [퐺퐹 (푝푛^2 ):퐺퐹(푝)][퐺퐹(푝):퐺퐹(푞)].
(28)Expansion order of algebraic number field퐺퐹(푞)is
푄 1 =[퐺퐹(푝푛^1 ) : 퐺퐹 (푞)] = 푛 1 ∗[
푝−1
푞
],
푄 2 =[퐺퐹(푝푛^2 ) : 퐺퐹 (푞)] = 푛 2 ∗[
푝−1
푞
].
(29)
Remove the noise terms퐿(0)and퐺(0)to get coefficients
information of homogeneous constant coefficient linear dif-
ferential equation.
3.3. Dynamic Game Model
Definition 7.The Computable complete and perfect informa-
tion dynamic game휏=[푃,푇,퐴,푆,푅,퐻,퐼,푂,푈]satisfies:
Participants are noted as 푃=
{Simulator,푃푖}(Simulator represents the nature
and parameter distributor).
The set of Types is푇={푇푖}(푇푖∈{honesty,fraud}).Actions set is퐴={퐴푖}(퐴푖∈{honesty,fraud}).
Strategy set is푆={푆푖}휑 : (푇푖,퐻푖,퐼푖,퐴푖)→푆푖.Rounds set is푅 < 푂(휆) ∧ 푅 ∈ 푍+.Full history set퐻={ℎ|ℎ=⨁푘푖=1퐴푖}(푖∈푅∧0≤
푘≤푅)isdepictedasgametree,whoserootisempty
history node 0.The information set퐼={퐼푖}canbetestedandis
perfect.Outcome set is푂={푂푖}훾 : (퐴푖,푆푖)→푂푖.
Utility function set is 푈={푈푖}훾 ∘ 휑 :
(푇푖,퐻푖,퐼푖,퐴푖,푆푖,푂푖)→푈푖and satisfies휕^2 푈푖<0.
The above game휏canbecalculatedinpolynomial
time.Definition 8.Computable complete and perfect information
dynamic game with푡 1 +푡 2 elastic equilibrium will reach the
equilibrium results, under the conditions that it satisfies the
Definition 7 and that each participants is rational. That is,
푈(휎푖,휎−푖)<푈(휎∗푖,휎−푖),휎is multiple real variable function
휎:(푇푖,퐻푖,퐼푖,퐴푖,푆푖,푂푖,푈푖)→푈(휎푖,휎−푖).
Theorem 9.The model converges to computable complete
and perfect information dynamic game with푡 1 +푡 2 elastic
equilibrium.Proof.Participants who accord with threshold signature con-
ditions possess superiority of Pr=휀(0<휀<1).Theycanget
threshold signature private key without the normal operation
of the model. Definitions of utility functions are as follows:
푈(0,푖)++: participants’ ideal utility without the normal oper-
ationofthemodeltoobtainthethresholdsignatureprivate
key;
푈(푟,푖)+ (0≤푟≤푟∗):theutilitythatparticipant푖gets
signature private key and others do not get it in푟round;
푈(푟,푖)− (0≤푟≤푟∗):utilitythatparticipant푖does not
comply with the normal execution of the model when model
run푟round;
푈(푟,푖)(0≤푟≤푟∗): utility that participant푖complies with
the normal execution of the model when model run푟round;
푈(푟∗,푖): normal utility that participant푖always complies
with the operation of the model obtains threshold signature
private key when model reaches the last one round;
푈(푟,−all)(0≤푟≤푟∗): utility that all participants do not
obtain the threshold signature private key. Illustrate that there
are some participants had deceived cause model abnormal
termination.
Utility function satisfies the strong partial:푈++(0,푖)>푈(푟,푖)+ >
푈(푟∗,푖)>푈(푟,−all).
Define events as follows.A: participant uses the advantage of Pr=휀(0<휀<
1)to crack threshold signature private key.B: participant implements protocol.C: participant takes honesty policy in round푟.D: participant takes fraud policy in round푟.We denote the utility of departing from the protocol as
푈exceptionand denote the expected utility as퐸(푈exception).We
can get the equation as follows.푈exception=휀푈(Pr(퐴))+(1−휀)푈(Pr(퐵)),푈(Pr(퐵))=푈(Pr(퐵|퐶)Pr(퐶)+Pr(퐵|퐷)Pr(퐷))=휆푈(푟∗,푖)+(1−휆)
푟
∑
푖=1