Task $1$: Generating Public and Private Keys

You are expected to write a function RSA$\_$Gen that accepts two prime numbers $p$ and $q$ at its input, and gives us three integer numbers N$,$ d$,$ and e at its output.

For $N=pq$ and $z=(p-1)(q-1),d$ and e must satisfy the following conditions:

zzzded$-1z,-1=0$modzN,$eN,dN,ec=RS{A}_{E}nc(N,e,m)$;$c=m^e$modNmc$(ab)modN=(amodN)(bmodN)modNN,dm=RS{A}_{D}ec(N,d,c)$;$m=c^d$modNcdm$(ab)modN=(amodN)(bmodN)1$

$e$ and $z$ have $no$ common factors, that $is,if$ you factor $z$ into its prime$-$number constituents, $e$ would not $be$ divisible $by$ any $of$ those prime numbers. Equivalently, the greatest common divisor $ofe$ and $z$ must $be$ one.

Choose $d$ such that $ed-1is$ divisible $byz,$ that $is,,ed-1=0(in$ modz$).$

Your public key $is$ then given $by$ the pair $(N,e),$ and your private key $is$ given $by(N,d).$

$If$ you are using MATLAB, these built$-in$ functions may help you $in$ this project: rem, mod, $gcd,$ and factor. Make yourself familiar with them and their $limitations.Do$ they correctly $do$ their job for very large numbers?

Task $2$: RSA Encryption

You are expected $to$ write a function $RS{A}_{E}nc$ that accepts the public key $(N,e)$ along with a message $m($ a natural number less than $N)at$ its input, and gives $us$ the encrypted version $ofmat$ its output:

$c=RS{A}_{E}nc(N,e,m)$;

You basically need $to$ write a code that calculates

$c=m^e$modN.

$Ifmore$ are large numbers, you may need $to$ calculate $cin$ multiple steps; remember the $limitationsof$ the MATLAB functions that you use and that $(ab)modN=(amodN)(bmodN)modN.$ Check out the supplementary video clip $on$ RSA protocol $to$ get some idea.

Task $3$: RSA Decryption

You are expected $to$ write a function $RS{A}_{D}ec$ that accepts the private key $(N,d)$ along with a ciphertext $c(a$ natural number less than $N)at$ its input, and gives $us$ the plaintext $mat$ its output:

$m=RS{A}_{D}ec(N,d,c)$;

You basically need $to$ write a code that calculates

$m=c^d$modN.

Again, $Ifcord$ are large numbers, you may need $to$ calculate $min$ multiple steps; remember the $limitationsof$ the MATLAB functions that you use and that $(ab)modN=(amodN)(bmodN).$ Check out the supplementary video clip $on$ RSA protocol $to$ get some idea.