A generalization of the Caesar cipher, known as the affine Caesar cipher, has th

Question

A generalization of the Caesar cipher, known as the affine Caesar cipher, has the following form: For each plaintext letter , substitute the ciphertext letter :

C = E([a, b], p) = (ap + b) mod 26

A basic requirement of any encryption algorithm is that it be one-to-one. That is, if p q, then E(k, p) E(k, q) . Otherwise, decryption is impossible, because more than one plaintext character maps into the same ciphertext character. The affine Caesar cipher is not one-to-one for all values of a. For example, for a = 2 b = 3, then E([a, b], 0) = E([a, b], 13) = 3.
a. Are there any limitations on the value of ? Explain why or why not.
b. Determine which values of are not allowed.
c. Provide a general statement of which values of are and are not allowed. Justify
your statement.

Explanation / Answer

a)

No, yet the successful scope of b is from 0 to 25. Since any number of b bigger than

25, we generally can supplant it with one from 0 to 25 to take the same capacity.

b)

0,2,4,6,8,10,12,13,14,16,18,20,22,24 are not permitted in the extent from 0 to 25. Like

the compelling scope of b is from 0 to 25, we additionally don't have to take any number bigger than 25.

c)

The estimation of an and 26 must be generally prime. From the inquiry we realize that

we should fulfill E(p) 6= E(q) if p 6= q, so we can derive from the suppose that if E(p) = E(q)whenp 6= q. (ap+b)mod26 = (aq+b)mod26 a(pq) = (k1k2)26.

At that point we can deduct that and ought to have no regular positive whole number element other

than 1.

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