P$4)$ In the ElGamal digital signature scheme, Alice selects a random number

k

such that

gcd$($k$,$p$1)=1$

$,$ then computes

r$\backslash$alpha

k

$($modp$)$

and

s$$k

$1$

$($m$$ar$)($modp$1)$

where

a

is the private key and

$($p$,\backslash$alpha $,\backslash$beta $)$

are public parameters in which

p

is a prime number and

$\backslash$beta $\backslash$alpha

a

$($modp$)$

$.$ There are many variations to this scheme that can be obtained by altering the signing equation. Here are some variations: $($a$)$ Consider the signing equation

s$$a

$1$

$($m$$kr$)($modp$1)$

$.$ Show that the verification

$\backslash$alpha

m

$(\backslash$alpha

a

$)$

s

r

r

$($modp$)$

is a valid verification procedure. $($b$)$ Consider the signing equation

s$$am$+$kr$($modp$1)$

$.$ Show that the verification

$\backslash$alpha

s

$(\backslash$alpha

a

$)$

m

r

r

$($modp$)$

is a valid verification procedure.P$4)$ In the ElGamal digital signature scheme, Alice selects a random number

k

such that

gcd$($k$,$p$1)=1$

$,$ then computes

r$\backslash$alpha

k

$($modp$)$

and

s$$k

$1$

$($m$$ar$)($modp$1)$

where

a

is the private key and

$($p$,\backslash$alpha $,\backslash$beta $)$

are public parameters in which

p

is a prime number and

$\backslash$beta $\backslash$alpha

a

$($modp$)$

$.$ There are many variations to this scheme that can be obtained by altering the signing equation. Here are some variations: $($a$)$ Consider the signing equation

s$$a

$1$

$($m$$kr$)($modp$1)$

$.$ Show that the verification

$\backslash$alpha

m

$(\backslash$alpha

a

$)$

s

r

r

$($modp$)$

is a valid verification procedure. $($b$)$ Consider the signing equation

s$$am$+$kr$($modp$1)$

$.$ Show that the verification

$\backslash$alpha

s

$(\backslash$alpha

a

$)$

m

r

r

$($modp$)$

is a valid verification procedure.