Encoding Information using Polynomials over Finite Fields

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Encoding Information using Polynomials over
Finite Fields

• Kevin Harness

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A Fundamental Transformation
(1,4)? (2,1)? (3,2)? (4,7)?
4 1 2 7
2 -9 11
y 2(x2)-9(x)11
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Relation to the Fourier Transform

• In some cases it is mathematically equivalent to
  the Finite Fourier Transform
• Fast Fourier Transform algorithms may apply
• Convolution Theorem may apply

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Existing Applications

• Reed-Solomon Codes
• Error correcting codes
• Works by sampling extra points on the curve
• Used for CDs, DVDs, some barcodes, etc.
• Shamir's Secret Sharing
• Splits a secret into parts
• Works by splitting up the points
• Used as a cryptographic building block

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Finite Fields

• A finite field consists of
• A set of elements that behave like numbers
• An addition operation ()?
• A multiplication operation ()?
• The operations must follow certain rules
• Allows subtraction and division
• Ensures that and behave as you expect

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Finite Field Sizes

• All finite fields have size pn for some prime p
• There exists a finite field of every such size
• All finite fields of a given size are isomorphic
• The finite field of size pn is called GF(pn)?

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Working with N-bit data

• We have N bits for each coefficient
• One option use GF(p1) for 2N lt p lt 2N1
• Arithmetic in GF(p1) is straightforward
• Outputs can require N1 bits
• Another option use GF(2N)?
• Arithmetic can be more complicated
• Outputs are exactly N bits

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Room for Improvement

• Protecting Secret Keys with Personal Entropy
• Describes a system for protecting data with a set
  of personal questions
• Uses Shamir's Secret Sharing Algorithm with a
  prime-sized finite field
• Requires workarounds to ensure that N-bit inputs
  are securely converted into N-bit outputs
• Could be simplified by using GF(2N)?