Internet Security - Dang Thanh Binh's Page

110 INTERNET SECURITY
Example 3.22 Compute (57) • (13) = (fe)
(57) = (01010111)
(57) • (02) = xtime(57) = (10101110) = (ae)
(57) • (04) = xtime(ae) = (01011100) ⊕ (00011011)
= (01000111) = (47)
(57) • (08) = xtime(47) = (10001110) = (8e)
(57) • (10) = xtime(8e) = (00011100) ⊕ (00011011) = (07)
Thus, it follows that
(57) • (13) = (57) •{(01) ⊕ (02) ⊕ (10)}
= (57) ⊕ (57) • (02) ⊕ (57) • (10)
= (57) ⊕ (ae) ⊕ (07)
= (01010111) ⊕ (10101110) ⊕ (00000111)
= (11111110) = (fe)
Polynomials with finite field elements in GF(2 8 )
A polynomial a(x) with byte-coefficient in GF(2 8 ) can be expressed in word form as:
a(x) = a3x 3 + a2x 2 + a1x + a0 ⇔ a = (a0,a1,a2,a3)
To illustrate the addition and multiplication operations, let
b(x) = b3x 3 + b2x 2 + b1x + b0 ⇔ b = (b0,b1,b2,b3)
be a second polynomial.
Addition is performed by adding the finite field coefficients of like powers of x
such that
a(x) + b(x) = (a3 ⊕ b3)x 3 + (a2 ⊕ b2)x 2 + (a1 ⊕ b1)x + (a0 ⊕ b0)
This addition corresponds to an XOR operation between the corresponding bytes in each
of the words. Multiplication is achieved as shown below:
The polynomial product c(x) = a(x) • b(x) is expanded and like powers are collected
to give
c(x) = a(x) • b(x) = c6x 6 + c5x 5 + c4x 4 + c3x 3 + c2x 2 + c1x + c0
= (c6,c5,c4,c3,c2,c1,c0)
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