Alice and Bob use elliptic curve encryption schema to communicate, and use the elliptic curve digital signature schema to sign messages.

Alice's public encryption key is the following

Elliptic Curve Group $y^2=x^3+Ax+$Bmodp

$A=8722749729$

$B=0$

$p=3927849327489732742098301298309218389728937482917$

$P_A=[3724,602906342049343611820419675309178006818745194654]$

$,$

$1734760601653880308873616205303816003016007523211$

Tolerance parameter $T=45.$

Bob's public encryption key is the following

Elliptic Curve Group $y^2=x^3+Ax+$Bmodp

$A=8722749729$

$B=0$

$p=3927849327489732742098301298309218389728937482917$

$P_B=[55,2552022818715546594530165777382068684546058213955]$

$,3151459300414441461488478318014699430983757085092$

Tolerance parameter $T=45.$

Alice's public signature key is the following

Elliptic Curve Group $y^2=x^3+Ax+$Bmodp

$A=8722749729$

$B=0$

$p=3927849327489732742098301298309218389728937482917$

$g_A=[723987,451322016145643867317671561270678778759571119857]$

$,$

$2450562033438136266241170969971653515326619594969]$

Bob's public signature key is the following

Elliptic Curve Group $y^2=x^3+Ax+$Bmodp

$]$

$[1594428288700492864201220211353219822695138388144$

One of the two sent a message to the other, with a digital signature attached. The padded ASCII version of the message has two packets:

packet $1$: $178197216205211210197208132184214201197215217$

packet $2$: $214201$

The signature on packet $1$ is

$y=322843805478469497496100778039428611308896099408$

$s=412986972317353396003644795415107659808384730639$

The signature on packet $2$ is

$y=322843805478469497496100778039428611308896099408$

$s=427696309917282746618353498255008340660091485337$

One of the two sent an encrypted message to the other. The encrypted message $($ciphertext$)$ is represented by two pairs of points $($one pair for each encrypted packet$)$:

The encryption version of packet $1$ is $[3392250666220429748352057521249133303617757145558,1939150282495281402743909945080558858688999245361],[1050636622523869006662499680251539236718340080433,2424946592070024418655054514705348236967025679428]$

The encryption version of packet $2$ is $3392250666220429748352057521249133303617757145558,1939150282495281402743909945080558858688999245361,[1988696510411214582210254167304130198728973460774,1222467321388159168322987359117953287886789562987]$

One of them by accident published also his$/$her private encryption key: $8237628684.$ Determine

$($a$)$ For whom it was encrypted i$.$e$.$ decrypt the message.

$($b$)$ By whom it was signed i$.$e verify the signature.