Eight friends decided to encrypt all their electronic communication. For encryption they

use the Elliptic Curve version of the system. For digital signatures they use the Elliptic

Curve version of the signature schema. Here are their public keys, listed by key$-$owner.

The following notation is used to denote the encryption and signature keys.

$1.$elliptic curve encryption keys:

$$GEG: The elliptic curve group.

$$gEG: a point in the group.

$$bEG$=$ ECTimes$($gEG$,$privatekey,GEG$).$

$$tEG: tolerance parameter.

$2.$elliptic curve digital signature keys:

$$GEGDS: The elliptic curve group.

$$gEGDS: a point in the group.

$$bEGDS$=$ ECTimes$($gEGDS$,$privatekey,GEGDS$).$

Alice

Elliptic Curve Public Key

GroupEG$=[12345,0,5502195571209402819093526123235067612330193079191439354976776763071]$

gEG$=[5,5091202217774727958900528120262884549746462763308570103342431790933]$

bEG$=[5169048018606356809083922735322604316989630212793858668477370341555,3576214157366831810350709448436036676384230817405358051409331533325]$

tEG$=45$

Bob

Digital Signature Public Key

GroupEG$=[12345,0,10063795724356324154858159842324666145419991393471217522376739]$

gEG$=[1,6903975208701294315389475241862116724280324639987271444108569]$

bEG$=[6721600486473718935723812395915971227892147738560323260170509,9580492932521114866404420461454292531271935647173642621179906]$

tEG$=45$

Carl

Elliptic Curve Public Key

GroupE$=[12345,0,5524160401955426921980756357667276500651389067901314535466798955199]$

gEG$=[1,1479912481638573783832793945604499019775622974861637792726871515254]$

bEG$=[170839132978393874134534251035410271871763701800195422951658139071,2499940854453627734776731919886101483095593486503452404086875260360]$

tEG$=45$

Dave

Elliptic Curve Public Key

GroupEG$=[12345,0,5579263905383263783681165872583736796925305797545734964609471660799]$

gEG$=[1,4072911192123752529465182838380536821095781828663305082706201664558]$

bEG$=[538198700432626268207047868190729336582752199213441903843798117545,5095577576367521615910358153120862012499541829274199337163343423938]$

tEG$=45$

Elvis

Elliptic Curve Public Key

GroupEG$=[12345,0,5735008556858309190508050373502955000071316000360078998382846148607]$

gEG$=[2,107276840688438889376400950314223507301646381468434976803711275319]$

bEG$=[2818405735632987736137542240049624419008253840010647134489795092478,4510709858380270913537135026156577808410352883844887079916106966096]$

tEG$=45$

Francis

Elliptic Curve Public Key

GroupEG$=[12345,0,5785524659442463851935386093815327844881237925340553260285477491247]$

gEG