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Pylonimial arithmetic in modulo 2

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• CRC: a more robust (error detection) check summing technique
  • Cyclic Redundancy Check (CRC):

    Cyclic Redundancy Check = a check "sum" computation technique based on division of: The message (bits)        and           A "CRC polynomial" The Cyclic Redundance Check (CRC) method is also known as polynomial code checksum              Wikipedia page: click here

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  • Facts on CRC:

    The Cyclic Redundancy Check (CRC) method is the most commonly used "check summing" method in use today When a communication professional use the term "check sum" he/she is referring to a CRC check sum

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    Why is CRC is so popular:

    CRC codes can detect common occuring errors that are caused by noise (lightning) in transmission channels. CRC can detect n consecutive bit errors CRC encoders/decoders are easy to construct: Electronic circuits used to compute the CRC code consists of handful of elementary gates and 1-bit memery circuits (D-flipflops) !!!

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• CRC computation
  • Fact:

    The CRC check sum computation uses: Polynomial arithmetic in modulo 2 arithmetic

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  • Furthermore:

    The polynomials will be represented using bit strings. Polynomial arithmetic operation becomes simple binary number operations using XOR !!!

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• Preliminary to CRC computation
  • We will discuss the follwoing material first:

    Polynomials representation in modulo 2 arithmetic Performing modulo 2 arithmetic on polynomials Representing polymonials using bit strings Performing modulo 2 arithmetic on polynomials represented in bit strings

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• Polynomial representation using modulo 2 arithmetic
  • Modulo 2 representation:

    An even coefficient is mapped to 0 An odd coefficient is mapped to 1

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  • Example representing polynomials in modulo 2:

    Polynomial: -x3 + 2x + 1 The polynomial in module 2 representation: x3 + 1 (mod 2) (Coefficient -1 is odd and is mapped to 1) (Coefficient 2 is even and is mapped to 0) (Coefficient 1 is odd and is mapped to 1)

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• Polynomial arithmetic using modulo 2 arithmetic
  • Example: addition

    (x3 + x) + (x + 1) = x3 + 2x + 1 = x3 + 1 (mod 2)

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  • Example: subtraction

    (x3 + x) - (x + 1) = x3 - 1 = x3 + 1 (mod 2)

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  • Example: multiplication

    (x2 + x) × (x + 1) = (x3 + x2) + (x2 + x) = x3 + 2x2 + x = x3 + x (mod 2)

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  • Example: division

    x2 + 1 ---------------- x + 1 / x3 + x2 + x x3 + x2 --------- x x + 1 ------ -1 x3 + x2 + x ------------ = x2 + 1 remainder 1 (mod 2) x + 1

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• Modulo 2 arithmetic
  • Summary: modulo 2 arithmetic

    0 1 0 1 + 0 + 0 + 1 + 1 ----- ----- ----- ----- 0 1 1 0

    Observation:

    The + (mod 2) operation is the same as an XOR (exclusive OR) operation used in logic !!!!

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• Polynomial representation using a bit string
  • Bit positions

    Bit string and position of a bit in the bit string: Bit string: 1010 ^^^^ |||| |||+-- bit position 0 ||+--- bit position 1 |+---- bit position 2 +----- bit position 3

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  • Bit string representation of a polynomial:

    The bit at position k is the coefficient of the factor xk in the polynomial

    Example:

    Polynomial: x5 + x4 + x2 + 1 x5 + x4 + x2 + 1 = 1x5 + 1x4 + 0x3 + 1x2 + 0x1 + 1x0 Bit string representation for the polynomial: 110101 ^^^^^^ |||||| |||||+----- x0 ||||+------ x1 |||+------- x2 ||+-------- x3 |+--------- x4 +---------- x5

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• Polynomial arithmetic using modulo 2 arithmetic with bit strings
  • Polynomial arithmetic in modulo 2 can be perform with bit string representation as follows:

    Replace the + (mod 2) operation with: XOR

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  • Example: addition

    (x3 + x) + (x + 1) = x3 + 2x + 1 = x3 + 1 (mod 2) In bit string representation: 1010 + 0011 ==> 1010 0011 ------ XOR 1001 ===> x3 + 1 (mod 2) Same result !!!

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  • Example: subtraction

    (x3 + x) - (x + 1) = x3 - 1 = x3 + 1 (mod 2) In bit string representation: 1010 + 0011 ==> 1010 0011 ------ XOR 1001 ===> x3 + 1 (mod 2) Same result !!!

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  • Example: multiplication

    (x2 + x) × (x + 1) = (x3 + x2) + (x2 + x) = x3 + 2x2 + x = x3 + x (mod 2) In bit string representation: 110 × 011 ==> 110 × 011 ----- 110 1100 00000 -------- XOR 1010 ====> x3 + x (mod 2) Same result !!!

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  • Example: division

    x2 + 1 ---------------- x + 1 / x3 + x2 + x x3 + x2 --------- x x + 1 ------ -1 x3 + x2 + x ------------ = x2 + 1 remainder 1 (mod 2) x + 1 In bit string representation: 101 ====> Quotient = x2 + 1 (mod 2) --------- 11 / 1110 11|| ---v| 01| 00| ----v 10 11 ----- 1 ====> remainder = 1 (mod 2)

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  • Conclusion:

    We can use the bit string representation an the XOR operation to perform the compution for the polynomial arithmetic