Alice, Bob and Carol use the the elliptic curves encryption scheme to communicate, and use the elliptic curve digital signature algorithm $($ECDSA$)$ to sign all messages they send. They decided on the following for their system:

For sending messages according to the elliptic curve encryption schema they will use the elliptic curve group $G=[32423423,0,111111756372541867431527].$ The $($base$)$ point relative to which the rest of their public key will be chosen for encryption purposes is $g$:$=[133213,42904742913851518927444].$

The public point $b,$ to be used for encryption, are as follows:

Alice: $[77012625572753817011635,107177820655281658062820]$

Bob: $[63567155336615021877739,36007178347937208049611]$

Carol:$[104225718616102818923406,48710727653703914714853]$

For signing messages according to the ECDSA they will use the elliptic curve $0,98766778,12357627687623542975694261$ and the base point $g=[2539662136869871783297850,1414894921445613162605806].$ It is known that the order of $g$ is $620176035713316419537.$

The public point $b,$ to be used for signature, are as follows:

Alice: $[5246859784562186496635824,5457612259949488195005309]$

Bob: $[5278780115068892215096216,6857888556017625502158912]$

Carol:$[11500587084934239979269346,4630177618473717713077113]$

The tolerance parameter for their system is $45.$ One of them sent a message to another member of the group, with a digital signature attached. The ciphertext is $[64822910729849959453845,54324913307643112006572]$ and $[40965317570056334177487,76589794174623293315486].$

The signature is $(979332919684899048601692,426149012637180793991).$

Determine

$($a$)$ Everyone's private keys.

$($b$)$ The message, for whom it was encrypted, and by whom it was signed.

$($c$)$ The random number used in signing the message.

$($d$)$ The random number used in encrypting the message.