Question $1.$ Given the prime $p$ from the data file for this homework, how many characters can we encode per message? Remember that the ASCII encoding of $k$ characters gives a number up to ${128}^k$ in size. You can take a base $b$ logarithm in Sage with the $log(x,b)$ command, usinglog$(x,b).n()$ to obtain a numerical estimate. $($You must justify your answer; saying 'I saw the number in the code' is not sufficient!$)$Alice publishes a public El Gamal key as the elliptic curve $E$ given in the

data file with points $P$ and $Q=kP,$ where $k$ is her secret key and is given in the data file, and cannot be determined from $Q$ without solving the elliptic curve discrete logarithm problem. Bob chooses a random ephemeral private key $l$ sends her the message encoded using the ECencode function as $P_m$ encrypted as $C_1,C_2$inE$(F_p)$ using the $El$ Gamal cipher: $C_1=lP,C_2=lQ+P_m.$

Notice Bob did not need $k$ to send Alice the message, only her public key was used.

The Given Code is as follows

def ECencode$($s$)$:

if len$($s$)>46$:

print$(\text{'}$Error: Trying to encode a message too long for this

curve.$\text{'})$

return

s $=$ str$($s$)$

x$0=$ sum$($ord$($s$[$i$])*128^$i for i in range$($len$($s$)))$

for i in range$(1000)$:

x $=$ mod$(1000*$x$0+$ i$,$ p$)$

y$2=$ x$^3+$a$*$x$+$b

if mod$($y$2,$p$)^(($p$-1)/2)==1$:

return E$([$x$,$y$2^(($p$+1)/4)])$

print$("$Failed to encode message on curve. This should literally almost

never happen."$)$

return False

def ECdecode$($P$)$:

n $=$ Integer$($P$[0])$

n $=($n$-$n$\%1000)//1000$

v $=[]$

while n $!=0$:

v$.$append$($chr$($n $\%128))$

n $//=128$

return $\text{'}\text{'}.$join$($v$)$

def prt$($x$,$ E$=\text{'}$E$\text{'})$:

if x$[2]==1$:

print$($str$($E$)+"("+$ str$($x$[0])+","+$ str$($x$[1])+")")$

else:

print$("$z$-$coordinate is not $1$: z $="+$ str$($x$[2]))$

p $=$

$32638362670776732582497808372513801812493719864404824108133552287033857635$

$957199184927795559383233731$

F $=$ GF$($p$)$

a $=$

$90512378278836473777257660031033108410833892402752231593075652989547873643$

$52962813346315403348732105$

b $=$

$28665754774437555325580892302508762109738382779922896285195927202693872891$

$275401767457645834312587697$

E $=$ EllipticCurve$($F$,[$a$,$ b$])$

P $=$

E$(268278132953802012296849398246005163783596010323274248180132058628857942$

$1418379319811693951113235402,$

$16869619464196207647323123436161130716215971876526267889484528178341832137$

$087769970754771506485262113)$

k $=692393314595748911633019465554228277654813344994872$

C$1=$

E$(299413739257553160568862108366943816092476424772845068755458341932386557$

$40439996735380855114886637859,$

$32003099420227818150640546832697309007650221568130300846699941139732213036$

$162076017115716729532301717)$

C$2=$

E$(291064601356432705969854254715660765199231967182750234611260584098351755$

$73196974958477478261532295650,$

$30435888933439216927938920653853078438467857258346541361541788686605413599$

$620043385425557980846296696)$

P$2=$

E$(258364748741314245007334219954446196732172476842279607735675783516371649$

$30549015595946119362968672437,$

$62679560412774440900540161772057211425708091098996771445077487298455966116$

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