Please solve it complete in 1 hour i will upvote you

(5) (a) Caesar's cipher is one of the simplest and most widely known encryption techniques, in which each letter in the original text in some alphabet is replaced by a letter some fixed number of positions down this alphabet. For example, in the standard alphabet {a,b, ..., z}, with a shift of 3, the letter a would be replaced by d, b would become e, and so on, the last three letters in the alphabet are replaced by a, b and c respectively. Describe formally (i.e., by means of a transition function) a Turing machine which encrypts (with shift 3) any word in the alphabet = {a,b,c,d,e). (b) Argue that the language {a"b"|n = 0,1,...} U{b"a"|n = 0, 1, ...} over the alphabet {a,b} belongs to the complexity class P. You don't need to give a formal but merely an implementation level description of the relevant Turing machine, i.e., show with sufficient detail why such a machine exists, and that its complexity is polynomial. (5) (c) Let L be the language of all equations = ao Xd +ajxd-1 + ... + ad-1X + ad = 0, ? with unknown X and integer coefficients an, Q1,...,0d-1, 2d, which have a solution in the integers. (i) Show that L is semi-decidable by describing, in general mathematical terms, an algorithm that takes an equation 20 Xd + a, xd-1 + ... +2d-1X+Qd = 0 with @o, 22, ..., Ad-1,20 Z as input and halts exactly when this equation has a solution in the integers. (5) (ii) Using the fact that L is actually decidable, describe an algorithm which for any equation 20xd + a28d-1 + ... +0d-1X + ad = 0 with ao, 22, ...,2d-1,00 Z, decides whether or not it has integer solutions, and if it does, finds at least one such solution. (5]