×
푟
∑
푖=1[
[
1
儨儨
儨儨儨퐺퐹(푝)∗
儨儨
儨儨儨
儨儨
儨儨儨퐺퐹(푝)∗
儨儨
儨儨儨
푈(+푟,푖)
+
(퐺퐹(푝)∗−1)
2儨儨儨
儨儨퐺퐹(푝)∗儨儨儨
儨儨
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
푈(−푟,all)]
],
푈exception=휀푈(++0,푖)+(1−휀)×[
[
푈(푟∗,푖)+(1−휆)
×
푟
∑
푖=1(
1
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
푈+(푟,푖)
+
(퐺퐹(푝)
∗
−1)2儨儨儨
儨儨퐺퐹(푝)∗儨儨儨
儨儨
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
푈(−푟,all))]
],
[퐺퐹(푝):퐺퐹(푞)]=
푝−1
푞
.
(30)
In our protocol,
푈exception<푈(푟∗,푖). (31)Distribution function satisfies푟∗=휓(휆). (32)The following formulas are met:휆<Φ∗[
[
푈(푟∗,푖)−휀푈(0,푖)++
1−휀
−
푟
∑
푖=1(
1
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
푈(푟,푖)+
+
(퐺퐹(푝)∗−1)
2儨儨
儨儨儨퐺퐹(푝)∗儨儨
儨儨儨
儨儨
儨儨儨퐺퐹(푝)∗
儨儨
儨儨儨
푈(−푟,all))]
],
(33)
in which
Φ=1×(푈(푟∗,푖)−
푟
∑
푖=1(
1
儨儨
儨儨儨퐺퐹(푝)∗
儨儨
儨儨儨
儨儨
儨儨儨퐺퐹(푝)∗
儨儨
儨儨儨
푈+(푟,푖)
+
(퐺퐹(푝)
∗
−1)2儨儨儨
儨儨퐺퐹(푝)∗儨儨儨
儨儨
儨儨儨
儨儨퐺퐹(푝)
∗儨儨儨
儨儨
푈−(푟,all)))−1.(34)
The above equation can determine the range of param-
eters selection, so that the model converges to computable
complete and perfect information dynamic game with푡 1 +푡 2
elastic equilibrium.
Theorem 10.The model can resist inner fraud.Proof.According to Theorem 9, a rational participant will
not depart from the protocol execution in any round. The
model overcomes the sensitivity of backward induction and
adopts mixed strategy equilibrium. If participants adopted
a deceptive strategy in the model execution of any round,
this caused the decrease in revenue of participants to푈−(푟,all).
When the protocol terminates, punishment strategies can be
used, thus putting an end to deceiving behavior effectively. So
the model can prevent inner fraud.4. Protocol Procedure
4.1. Parameters Generation Process.Determine the order of
set퐴and set퐵; determine the threshold value according to
the requirements, respectively. Select big prime푞, 푝meets
푞 | (푝 − 1). Select primitive element푔 1 in finite field퐺퐹(푝푛^1 )
and푔 2 in finite field퐺퐹(푝푛^2 ).Theparticipantsinset퐴and
set퐵select signature private key as the second component of
the Okamoto signature, respectively.
Parameter sequence generator generates coefficient con-
stants vector of homogeneous constant coefficient linear
differential equation:푎^0 =(푎^01 ,푎^02 ,...,푎푛−푡^01 )(푎푖^0 ∈푍푞),
푏^0 =(푏 10 ,푏 20 ,...,푏^0 푛−푡 1 )(푏^0 푖∈푍푞).
(35)
Superscript represents signature number of times; 0
represents the first signature.4.2. Dynamic Games Process.Rounds controller according
to Poisson distribution with parameter휆secret generates
threshold signature round푟∗. According to the number of
participants in set퐴and set퐵,thethresholdvaluegenerates
coefficient constants vector of polynomial퐺(푥)and퐿(푥),
respectively:푢^0 =(푢^01 ,푢 20 ,...,푢^0 푛 1 )(푢^0 푖∈푍푞푛 1 ),
푙^0 =(푙^01 ,푙^02 ,...,푙푛^02 )(푙^0 푖∈푍푞푛 2 ).
(36)
Superscript signature represents the number of rounds; 0
represents the first round.
Parameter distributor according to ( 17 )and( 22 )dis-
tributes and publicizes points. Participants in set퐴and set
퐵can use the verifiable parameter distribution module for
verification. If there is no cheating behavior, the protocol
continues to execute. Otherwise, the verifiable parameter
distribution module goes to the interrupt processing. In every
round of the games, the players in set퐴and set퐵use the
published points sequence and generate퐺(0)푟and퐿(0)푟,
respectively.