MATH 195: CRYPTOGRAPHY HOMEWORK #9

Problem F5. Verify that

X^{8} + X^{4} + X^{3} + X + 1

is irreducible in 𝔽_{2}[X]. Problem F6. Put f = X^{8} + X^{4} + X^{3} + X + 1 ∈ 𝔽_{2}[X], and let

a = 00001100 = X^{3} + X^{2} ∈ 𝔽_{2}[X]/(f).

Compute a^{5}. Can you find an explicit embedding of

𝔽_{2}[X]/(X^{4} + X^{3} + X^{2} + X + 1)

as a subfield in 𝔽_{2}[X]/(f)? Problem F7. Let p be prime. Prove that in 𝔽_{p}[X]/(X^{p} − X − 1) one has

X^{p^{i}} = X + (i mod p) = \underbrace{00 … 00}_{p−2} 1(i mod p)

for i any positive integer. Deduce that X^{p} − X − 1 is irreducible in 𝔽_{p}[X].

Date: April 2, 2002.

F5, F6, F7.