Components and Design Techniques for Digital Systems-Computer Security-Lecture Slides, Slides of Computer Security

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EECS 150 - Components and Design

Techniques for Digital Systems

Lec 26 – CRCs, LFSRs

(and a little power)

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Galois Fields - the theory behind LFSRs

• LFSR circuits performs multiplication on a field****.
• A field is defined as a set with the following: - two operations defined on it: » “addition” and “multiplication” - closed under these operations - associative and distributive laws hold - additive and multiplicative identity elements - additive inverse for every element - multiplicative inverse for every non-zero element - Example fields: - set of rational numbers - set of real numbers - set of integers is not a field (why?) - Finite fields are called Galois fields. - Example: - Binary numbers 0,1 with XOR as “addition” and AND as “multiplication”. - Called GF(2). - 0+1 = 1 - 1+1 = 0 - 0-1 =? - 1-1 =?

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So what about division (mod)

x^4 + x^2

x = x

(^3) + x with remainder 0

x^4 + x^2 + 1

X + 1

= x^3 + x^2 with remainder 1

X + 1 x^4 + 0x^3 + x^2 + 0x + 1

x^3

x^4 + x^3 x^3 + x^2

+ x^2

x^3 + x^2

0x^2 + 0x

+ 0x

0x + 1

+ 0

Remainder 1

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Polynomial division

• When MSB is zero, just

shift left, bringing in next

bit

• When MSB is 1, XOR with

divisor and shiftl

1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0

0 0 0 0

1 0 0 1 1 0 0 1 0 1

1

0 1 0 1 0

0

1 0 1 0 1 1 0 0 1 1

1

0 0 1 0 0

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Galois Fields – Primitives

^0 = 1 ^1 = x ^2 = x^2 ^3 = x^3 ^4 = x + 1 ^5 = x^2 + x ^6 = x^3 + x^2 ^7 = x^3 + x + 1 ^8 = x^2 + 1 ^9 = x^3 + x ^10 = x^2 + x + 1 ^11 = x^3 + x^2 + x ^12 = x^3 + x^2 + x + 1 ^13 = x^3 + x^2 + 1 ^14 = x^3 + 1 ^15 = 1

• Note this pattern of coefficients matches the bits from our 4-bit LFSR example.
• In general finding primitive polynomials is difficult. Most people just look them up in a table, such as:

^4 = x^4 mod x^4 + x + 1 = x^4 xor x^4 + x + 1 = x + 1

Multiplication by x  shift left Taking the result mod p(x)  XOR-ing with the coefficients of p(x) when the most significant coefficient is 1. Obtaining all 2 n-1 non-zero  Shifting and XOR-ing 2 n-1 times. elements by evaluating xk

• x^2 + x + Primitive Polynomials
• x^3 + x +
• x^4 + x +
• x^5 + x^2 +
• x^6 + x +
• x^7 + x^3 +
• x^8 + x^4 + x^3 + x^2 +
• x^9 + x^4 +
• x^10 + x^3 +
• x^11 + x^2 +
  • x^12 + x^6 + x^4 + x +
  • x^13 + x^4 + x^3 + x +
  • x^14 + x^10 + x^6 + x +
  • x^15 + x +
  • x^16 + x^12 + x^3 + x +
  • x^17 + x^3 +
  • x^18 + x^7 +
  • x^19 + x^5 + x^2 + x+
  • x^20 + x^3 +
  • x^21 + x^2 +
    • x^22 + x +
    • x^23 + x^5 +
    • x^24 + x^7 + x^2 + x +
    • x^25 + x^3 +
    • x^26 + x^6 + x^2 + x +
    • x^27 + x^5 + x^2 + x +
    • x^28 + x^3 +
    • x^29 + x +
    • x^30 + x^6 + x^4 + x +
    • x^31 + x^3 +
• Galois Field Hardware x 32 + x 7 + x 6 + x 2 +