Documentation.SBV.Examples.Misc.Polynomials

Simple usage of polynomials over GF(2^n), using Rijndael's finite field: http://en.wikipedia.org/wiki/Finite_field_arithmetic#Rijndael.27s_finite_field

The functions available are:

pMult

GF(2^n) Multiplication

pDiv

GF(2^n) Division

pMod

GF(2^n) Modulus

pDivMod

GF(2^n) Division/Modulus, packed together

Note that addition in GF(2^n) is simply xor, so no custom function is provided.

Documentation

type GF28 = SWord8

Helper synonym for representing GF(2^8); which are merely 8-bit unsigned words. Largest term in such a polynomial has degree 7.

gfMult :: GF28 -> GF28 -> GF28

Multiplication in Rijndael's field; usual polynomial multiplication followed by reduction by the irreducible polynomial. The irreducible used by Rijndael's field is the polynomial x^8 + x^4 + x^3 + x + 1, which we write by giving it's exponents in SBV. See: http://en.wikipedia.org/wiki/Finite_field_arithmetic#Rijndael.27s_finite_field. Note that the irreducible itself is not in GF28! It has a degree of 8.

NB. You can use the showPoly function to print polynomials nicely, as a mathematician would write.

multUnit :: GF28 -> SBool

States that the unit polynomial 1, is the unit element

multComm :: GF28 -> GF28 -> SBool

States that multiplication is commutative

multAssoc :: GF28 -> GF28 -> GF28 -> SBool

States that multiplication is associative, note that associativity proofs are notoriously hard for SAT/SMT solvers

polyDivMod :: GF28 -> GF28 -> SBool

States that the usual multiplication rule holds over GF(2^n) polynomials Checks:

   if (a, b) = x pDivMod y then x = y pMult a + b

being careful about y = 0. When divisor is 0, then quotient is defined to be 0 and the remainder is the numerator. (Note that addition is simply xor in GF(2^8).)

testGF28 :: IO ()

Queries