Slide $6$

Direct proof:

Construct adversary A as follows:

Mo $=(0\text{'},0)$and M$,=(0\text{'},1)$

Challenge ciphertext C $=($G$,)$

Adversary A: if $4=$C$,$then b$\text{'}=0$else b$\text{'}=1$

If b$=0$then b$\text{'}=0$

If b$=1$then b$\text{'}=1$

So b$=$b$\text{'}$with probability $1$

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THEOREM $3\mathrm{.}20$

If Enc is deterministic and stateless the encryption scheme cannot satisfy definition $3\mathrm{.}18$

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Slide $3$

The multiple$-$message eavesdropping experiment PrivKA, $($n$)$:

$$The adversary A is given input $1",$and outputs a pair of

$$equal$-$length lists of messages Mo $=($mo$,1,...,$mo$,$t$)$and M$1=$

$($M$1,1,...,$M$1,$t$),$with mo$,$i $=$m$1,$il for all i$.$

$$A key k is generated by running Gen$(1"),$and a uniform bit bE $\{0,1\}$is chosen. For all i$,$the ciphertext ci $+$Ench $($mb$,$i$)$is computed and the list C$\text{'}=($C$1,..,$Ct$)$is given to A$.$

$$A outputs a bit b$\text{'}.$

$$The output of the experiment is defined to be $1$if $6\text{'}=$b$,$and

$$O otherwise.

DEFINITION $3\mathrm{.}18$A private$-$key encryption scheme II $=($Gen$,$Enc$,$ Dec$)$

has indistinguishable multiple encryptions in the presence of an eavesdropper if for all probabilistic polynomial$-$time adversaries A there is a negligible function negl such that

Pr PrivKAH$($n$)=1<=$

$-+$negl$($n$).$

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Question $1$

Consider the following statement: Theorem $3\mathrm{.}20($slide $6)$establishes that a necessary and sufficient condition for an encryption scheme $$to satisfy Definition $3\mathrm{.}18($slide $3)$is that the Enc algorithm of $$be randomized or stateful. Discuss.

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Slide $8$

Suppose $0=$Fk$,$for any k

Then D will always output $1$

Pr$|$DFK $()(10)=1]=1$

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Question $2$

In Example $3\mathrm{.}5($slide $8)$we have the following claim

Suppose O $=$Fk $,$for any k

$$ Then D will always output $1$

Explain why this claim is true.

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Slide $8$

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The CPA indistinguishability experiment PrivKA,$($n$)$:

$4,($n$)$:

$$A key k is generated by running Gen$(1").$

$$The adversary A is given input $1"$and oracle access to Enck$(),$and outputs a pair of messages mo$,$my of the same length.

$$A uniform bit b e $10,1\}$is chosen, and then a ciphertext c $<$Enck$($mb$)$is computed and given to A$.$

$$The adversary A continues to have oracle access to Enck$(),$and outputs a bit $6\text{'}.$

$$The output of the experiment is defined to be $1$if b$\text{'}=$b$,$and

$$O otherwise. In the former case, we say that A succeeds.

DEFINITION $3\mathrm{.}21$A private$-$key encryption scheme II $=($Gen$,$Enc$,$ Dec$)$

has indistinguishable encryptions under a chosen$-$plaintext attack, or is CPA$-$secure$,$if for all probabilistic polynomial$-$time adversaries A there is a negligible function negl such that

Pr PrivKAa$($n$)=1<=$

$1$

$=+$negl$($n$),$

$2$

where the probability is taken over the randomness used by A$,$as well as the randomness used in the experiment.

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Let q$($n$)$be a polynomial upper bound on number of queries A makes to the oracle

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Question $3$

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Slide $8$ introduces the notation $$Let q$($n$)$be a polynomial upper bound on number of queries A makes to the oracle.$$Does a PPT adversary A satisfy this requirement? Explain.

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Slide $4$

CONSTRUCTION $3\mathrm{.}28$

Let F be a pseudorandom function. Define a fixed$-$length, private$-$key encryption scheme for messages of length n as follows:

$$Gen: on input $1",$choose uniform k $\{0,1\}"$and output it$.$

$$Enc: on input a key k $\{0,1\}"$and a message m $\{0,1\}",$choose uniform r E $\{0,1\}"$and output the ciphertext

c:$=($r$,$Fk$($r$)$em$).$

$$Dec: on input a key k E $\{0,1\}"$and a ciphertext c $=($r$,$s$),$output the message

m:$=$Fk$($r$)$@s$.$

A CPA$-$secure encryption scheme from any pseudorandom function.

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Question $4$

In context of construction $3\mathrm{.}28($slide $4)$let n$=128$so the encryption scheme can be used to encrypt a message of length $128$bits$.$ How many possible ciphertexts exist for a single plaintext message and a fixed key k$?$Explain$.$

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$($Plz ans by $1,2,3,4$thank you $)$

$($Plz do not use chat gpt $)$