Implement the hash functions given $in(a)$ and $(b)$:

$(a)$ Given $n=pqbe$ the product $of$ two distinct large primes and let $h1(x)be$ a hash function, where $h1(x)=x2(modn).$ Attacker knows $n,$ but not $p$ and $q.$

$(b)$ Suppose a message $mis$ divided into blocks $of$ length $160$ bits: $m=M1||M2||....||Mr.$ Let $h2(x)be$ a hash function, where $h2(m)=M1M2...Mn.$

For each $of$ the hash functions, analyze $on$ paper whether they adhere $to$ the following properties: one$-$way, weak collision resistance, and strong collision resistance. $If$ any function fails $to$ uphold any $of$ these properties, develop a program $to$ illustrate $it.$ Specifically:

One$-$way: $If$ the function doesn't comply with the one$-$way property, create a program $to$ compute $x$ given its hash value $h(x).$ Consequently, demonstrate that your code can calculate $at$ least one such $x.$

Weak$-$collision Resistance: Should the function lack weak$-$collision resistance, construct a program $to$ discover $ay$ given $x$ such that $h(x)=h(y).$ Subsequently, demonstrate that your code can find $at$ least one such $y.$

Strong$-$collision Resistance: $If$ the function fails $to$ satisfy strong$-$collision resistance, devise a program $to$ identify a pair $(x,y)$ such that $h(x)=h(y).$ Then, showcase that your code can find $at$ least one such pair $(x,y).$

Question $1(60$ points$)$:

Implement the hash functions given $in(a)$ and $(b)$:

$(a)$ Given $n=pqbe$ the product $of$ two distinct large primes and let $h_1(x)be$ a hash function, where

$h_1(x)=x^2(modn).$ Attacker knows $n,$ but not $p$ and $q.$

$(b)$ Suppose a message $mis$ divided into blocks $of$ length $160$ bits: $m=M_1||M_2||dots..||M_r.$ Let $h_2(x)be$

a hash function, where $h_2(m)=M_{1}o+M_2$dotso$+M_n.$

For each $of$ the hash functions, analyze $on$ paper whether they adhere $to$ the following properties:

one$-$way, weak collision resistance, and strong collision resistance. $If$ any function fails $to$ uphold any

$of$ these properties, develop a program $to$ illustrate $it.$ Specifically:

One$-$way: $If$ the function doesn't comply with the one$-$way property, create a program $to$ compute $x$

given its hash value $h(x).$ Consequently, demonstrate that your code can calculate $at$ least one such

$x.$

Weak$-$collision Resistance: Should the function lack weak$-$collision resistance, construct a program

$to$ discover $ay$ given $x$ such that $h(x)=h(y).$ Subsequently, demonstrate that you Pls remember that attacker know nothing about b