1. Suppose we have a set of blocks encoded with the RSA algorithm and we don’t have the private key. Assume  is the public key. Suppose also someone tells us he or she knows one of the plaintext blocks has a common factor with n. Does this help us in any way?

2. Show how RSA can be represented by matrices M1, M2, and M3 of Problem 21.4

Problem 21.4

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  by a vector M1 of length N, in which the kth entry is the value of Æ’₁(k) Similarly Æ’₂ and Æ’₃ 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:

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:

Convention: The ith element of M1 corresponds to  The ith row of M2 corresponds to  the jth column of M2 corresponds to  The ith row of M3 corresponds to  the jth column of M3 corresponds to  

b. Describe the use of this set of tables to perform encryption and decryption between two users

c. Argue that this is a secure scheme.