sbv-0.9.8: Symbolic Bit Vectors: Prove bit-precise program properties using SMT solvers.




Simple usage of polynomials over GF(2^n), using Rijndael's finite field:

The functions available are:

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

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



type GF28 = SWord8Source

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

(<*>) :: GF28 -> GF28 -> GF28Source

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: 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 -> SBoolSource

States that the unit polynomial 1, is the unit element

multComm :: GF28 -> GF28 -> SBoolSource

States that multiplication is commutative

multAssoc :: GF28 -> GF28 -> GF28 -> SBoolSource

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

polyDivMod :: GF28 -> GF28 -> SBoolSource

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 ()Source