Question $1($Marks: $15)-$ Cryptographic Data Objects

B has just received the following message, which represents a cryptographic data object:

$\{($

$\{($KPbB$)$KPrS mod KPbS$\}$K$1,$

$\{|($NB$,$ NA$,\{\{(\{$K$2\}$KPbB$,$ NS$)\}($G$1)$KPrA mod NA$\}$K$1,\{|\{(\{$G$3\}($KPbA$)$KPrS mod KPbS$,$

G$2)\}$K$1|\}$KPrB$)|\}$KPrA

$)\}$KBS

The following explains various terms in this object and some of the abbreviations used:

$\{$M$\}$K represents the encryption of some message$/$data M using the key K

$\{|$M$|\}$K represents the digital signing of some message$/$data M using the key K $$ NX represents a nonce $($i$.$e$.$ a fresh and possibly random number used once only$)$ generated by X

$$ KpbX represents the public part of the key pair presumably owned by X

$$ KprX represents the private part of the key pair presumably owned by X

$$ KAB represents a symmetric key shared between A and B

$$ K $($or K$1,$ K$2,$ K$3$ etc.$)$ represents some arbitrary key with no assumptions about its scope

$$ M represents some alphanumeric$/$textual message with no assumptions $$ G$1,$ G$2,$ G$3$ etc. are prime numbers

which one of the following sets of keys, nonces, prime numbers, and alphanumeric$/$textual messages $$best$$ represents B$$s knowledge, after B applies any number of possible cryptographic operations to the object above, and assuming that B already has access to key K$1$ and the public key of any agent:

a$)$ KBS $,$ G$2,$ KPrB

b$)($KPbB$)$KPrS mod KPbS$,$G$2,$KBS$,$ KPrB$,\{($KPbB$)$KPrS mod KPbS$\}$K$1,$ NA $,$ NB

c$)$ NA $,$ NB

d$)$ NA $,$ NB $,$ KBS $,$ KPrB

e$)\{($KPbB$)$KPrS mod KPbS$\}$K$1,\{|($NB$,$ NA$,\{\{(\{$K$2\}$KPbB$,$ NS$)\}($G$1)$KPrA mod

NA$\}$K$1,\{|\{(\{$G$3\}($KPbA$)$KPrS mod KPbS$,$ G$2)\}$K$1|\}$KPrB$)|\}$KPrA $,$ NA $,$ NB $,$ KBS

$,$ KPrB $,\{($KPbB$)$KPrS mod KPbS

f$)$ G$2,$ NA $,$ NB $,$ G$1,$ KBS $,$ KPrB

g$)($KPbB$)$KPrS mod KPbS $,$ NA $,$ NB $,$ G$2,$ KBS $,$ KPrB

h$)($KPbB$)$KPrS mod KPbS $,($G$1)$KPrA mod NA $,$ NA$,$ NB $,$ G$2,$ KBS $,$ KPrB

i$)($KPbB$)$KPrS mod KPbS $,$ G$3,$ G$2,$ KBS $,$ KPrB

j$)($KPbB$)$KPrS mod KPbS $,$ NA $,$ NB $,$ G$2,$ KBS $,$ KPrB $,$ G$3,($KPbA$)$KPrS mod KPbS

k$)$ NB