Let p = 227. The element alpha = 2 is primitive in Z_p*. (a) Compute alpha^32 alpha^40 alpha^59 and alpha^156 modulo p, and factor them over the factor base {2, 3, 5, 7, 11}. (b) Using the fact that log 2 = 1, compute log 3, log 5, log 7 and log 11 from the factorizations obtained above (all logarithms are discrete logarithms in Z_p* to the base alpha). (c) Now suppose we wish to compute log 173. Multiply 173 by the "random" value 2^177 mod p. Factor the result over the factor base, and proceed to compute log 173 using the previously computed logarithms of the numbers in the factor base