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Lecture 6 Notes

# Exactly once semancs at least once at most once at

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Unformatted text preview: n(x) = m(x)x3. Let C(x) = x3 + x2 + 1 –  Find q(x) and r(x) s.t. n(x) = q(x)C(x) + r(x) and degree of r(x) < degree of C(x) –  Analogous to taking 11 mod 5 = 1 hnp://en.wikipedia.org/wiki/Cyclic_redundancy_check Polynomial Division Example •  Just long division, but addi)on/subtrac)on is XOR Generator 11111001 1101 10011010000 1101 Message 1001 1101 1000 1101 1011 1101 1100 1101 1000 1101 101 Remainder CRC •  Select a divisor polynomial C(x), degree k –  C(x) should be irreducible – not expressible as a product of two lower- degree polynomials in Z2[x] •  Add k bits to message –  Let n(x) = m(x)xk (add k 0’s to m) –  Compute r(x) = n(x) mod C(x) –  Compute n(x) = n(x) – r(x) (will be divisible by C(x)) (subtrac)on is XOR, just set k lowest bits to r(x)!) •  Checking CRC is easy –  Reduce message by C(x), make sure remainder is 0 Why is this good? •  Suppose you send m(x), recipient gets m’(x) –  E(x) = m’(x) – m(x) (all the incorrect bits) –  If CRC passes, C(x) divides...
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