Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Transcribed Image Text:An irreducible polynomial of degree 8 over GF(2) is given as f(x) =x+x+x'+x +1. This can
be used to define GF(2)=GF(256) which is frequently used to design a code over the ASCII
symbols consisting of 8 bits.
We will use the following notation, for a, e GF(2),
e ar + a,r+ ar ++ aa +a,
(A) Write a computer program of addition and multiplication for all the binary 8-tuples as above
and answer to the following questions:
(B) Find any member a in GF(256) with the property that a, a², a²̟., a5 are all distinct,
Calculate as al28, and a4.
ANSWER:
) :a binary 8-tuple
):a binary 8-tuple
):a binary 8-tuple
):a binary 8-tuple
a = (
a4 = (
a12 =(
ass = (
(C) Calculate the multiplications a a, att . atza, and a al0 using your program in (A).
Verify, for example, that a at = al+64 = ats, a*. a1zi - att12- al92 etc.
(D) Define a linear function g from GF(256) to itself such that g(x) = x for any x in GF(256).
Here, the linearity implies that g preserves two operations of the field.
That is, for any x,y in GF(256), we have
g(x + y) = g(x) + g(y) and g(x-y) = g(x) - gV).
Either prove that g is 1-1 or find a counterexample that g is NOT 1-1.
Either prove that g is onto or find a counterexample that g is NOT onto.
Report must contain your program source code and the answers to the above three problems.
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