Question

6. What is the orbit of 2 in the group Z_7 under multiplication modulo 7? Is 2 a generator?

7. What is the residue of 101101 modulo 1101 using these as representations of polynomials with binary coefficients?

8. List all irreducible polynomials with binary coefficients of degree 4 or less. (Hint: produce a times table that shows the minimum number of products needed.) Show these as binary numbers (omitting the indeterminant) and as decimal numbers (interpreting the binary number into decimal). Is 23 a prime polynomial in this field?

9. Interpreting these decimal numbers into coefficients of polynomials with binary coefficients, what is the product of 11 and 10 modulo 31 in GF(2^4) over P = 31? (Hint: it is not 17.) Use polynomial multiplication and reduction by long division.

10. Find a generator g for GF(2^4) over P=31. Show that g is a generator for the field. Find the powers of g that produce 10 and 11 (as in question 9) and show that their product using the sum of the generator's powers gives the same results you got in problem 9.

Answer 1

these question are very tricky ..because it is based on number theory applications ... so am doind first two questions...

please post rest questions in next time ...

6)

7)

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