8.Prior to the discovery of any specific public-key schemes, such as RSA, an existence proof wasdeveloped whose purpose was to demonstrate that public-key encryption is possible in theory.Consider the functionsf₁(x₁) = Z₁; f₂(x2, y2) = Z2; f(x3, y3) = Z3where all values are integers with Function can be represented by a vector M1 of length N, in whichthe kth entry is the value of Similarly, and can be represented by matrices M2 and M3. The intent is torepresent the encryption/decryption process by table lookups for tables with very large values of N.Such tables would be impractically huge but could, in principle, be constructed. The scheme works asfollows: Construct M1 with a random permutation of all integers between 1 and N; that is, eachinteger appears exactly once in M1. Construct M2 so that each row contains a random permutation ofthe first N integers. Finally, fill in M3 to satisfy the following condition:fi(fe(fi(k),p),k) = p for all k, p with 1 ≤k, p≤NIn words,1. M1 takes an input k and produces an output x.2. M2 takes inputs x and p giving output z.3. M3 takes inputs z and k and produces p.The three tables, once constructed, are made public.(a)It should be clear that it is possible to construct M3 to satisfy the preceding condition.As an example, fill in M3 for the following simple case:54M1 = 235 2 3 44 2 53 23142 5 3 4 1M2 1134 525(c) Argue that this is a secure scheme.M3 =Convention: The ith element of M1 corresponds to "ki" The ith row of M2 corresponds to "x= i" the jth column of M2 corresponds to "p=j" The ith row of M3 corresponds to "z=i" thejth column of M3 corresponds to "k=j".[6 marks](b) Describe the use of this set of tables to perform encryption and decryption between two users.[3 marks][3 marks]

a.Finding value for the third matrix (M3):The matrix M3 elements can be attained using the below…

Prior to the discovery of any specific public-key schemes, such as RSA, an existence proof was developed whose purpose was to demonstrate that public-key encryption is possible in theory. Consider the functions f₁(x₁)=z₁; f₂(x2, y2)=Zz; f(x3, Y3) = Z3

Prior to the discovery of any specific public-key schemes, such as RSA, an existence proof was developed whose purpose was to demonstrate that public-key encryption is possible in theory. Consider the functions f₁(x₁)=z₁; f₂(x2, y2)=Zz; f(x3, Y3) = Z3

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Question

8.
Prior to the discovery of any specific public-key schemes, such as RSA, an existence proof was
developed whose purpose was to demonstrate that public-key encryption is possible in theory.
Consider the functions
f₁(x₁) = Z₁; f₂(x2, y2) = Z2; f(x3, y3) = Z3
where all values are integers with Function can be represented by a vector M1 of length N, in which
the kth entry is the value of Similarly, and can be represented by matrices M2 and M3. The intent is to
represent the encryption/decryption process by table lookups for tables with very large values of N.
Such tables would be impractically huge but could, in principle, be constructed. The scheme works as
follows: Construct M1 with a random permutation of all integers between 1 and N; that is, each
integer appears exactly once in M1. Construct M2 so that each row contains a random permutation of
the first N integers. Finally, fill in M3 to satisfy the following condition:
fi(fe(fi(k),p),k) = p for all k, p with 1 ≤k, p≤N
In words,
1. M1 takes an input k and produces an output x.
2. M2 takes inputs x and p giving output z.
3. M3 takes inputs z and k and produces p.
The three tables, once constructed, are made public.
(a)
It should be clear that it is possible to construct M3 to satisfy the preceding condition.
As an example, fill in M3 for the following simple case:
5
4
M1 = 2
3
5 2 3 4
4 2 5
3 2
314
2 5 3 4 1
M2 1
13
4 5
25
(c) Argue that this is a secure scheme.
M3 =
Convention: The ith element of M1 corresponds to "ki" The ith row of M2 corresponds to "x
= i" the jth column of M2 corresponds to "p=j" The ith row of M3 corresponds to "z=i" the
jth column of M3 corresponds to "k=j".
[6 marks]
(b) Describe the use of this set of tables to perform encryption and decryption between two users.
[3 marks]
[3 marks]

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