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Diffstat (limited to 'src/Arithmetic/Karatsuba.v')
-rw-r--r-- | src/Arithmetic/Karatsuba.v | 49 |
1 files changed, 49 insertions, 0 deletions
diff --git a/src/Arithmetic/Karatsuba.v b/src/Arithmetic/Karatsuba.v new file mode 100644 index 000000000..0f20bb238 --- /dev/null +++ b/src/Arithmetic/Karatsuba.v @@ -0,0 +1,49 @@ +Require Import Coq.ZArith.ZArith. +Require Import Crypto.Algebra.Nsatz. +Require Import Crypto.Util.ZUtil. +Local Open Scope Z_scope. + +Section Karatsuba. + Context {T : Type} (eval : T -> Z) + (sub : T -> T -> T) + (eval_sub : forall x y, eval (sub x y) = eval x - eval y) + (mul : T -> T -> T) + (eval_mul : forall x y, eval (mul x y) = eval x * eval y) + (add : T -> T -> T) + (eval_add : forall x y, eval (add x y) = eval x + eval y) + (scmul : Z -> T -> T) + (eval_scmul : forall c x, eval (scmul c x) = c * eval x) + (split : Z -> T -> T * T) + (eval_split : forall s x, s <> 0 -> eval (fst (split s x)) + s * (eval (snd (split s x))) = eval x) + . + + Definition karatsuba_mul s (x y : T) : T := + let xab := split s x in + let yab := split s y in + let xy0 := mul (fst xab) (fst yab) in + let xy2 := mul (snd xab) (snd yab) in + let xy1 := sub (mul (add (fst xab) (snd xab)) (add (fst yab) (snd yab))) (add xy2 xy0) in + add (add (scmul (s^2) xy2) (scmul s xy1)) xy0. + + Lemma eval_karatsuba_mul s x y (s_nonzero:s <> 0) : + eval (karatsuba_mul s x y) = eval x * eval y. + Proof using Type*. cbv [karatsuba_mul]; repeat rewrite ?eval_sub, ?eval_mul, ?eval_add, ?eval_scmul. + rewrite <-(eval_split s x), <-(eval_split s y) by assumption; ring. Qed. + + + Definition goldilocks_mul s (xs ys : T) : T := + let a_b := split s xs in + let c_d := split s ys in + let ac := mul (fst a_b) (fst c_d) in + (add (add ac (mul (snd a_b) (snd c_d))) + (scmul s (sub (mul (add (fst a_b) (snd a_b)) (add (fst c_d) (snd c_d))) ac))). + + Local Existing Instances Z.equiv_modulo_Reflexive RelationClasses.eq_Reflexive Z.equiv_modulo_Symmetric Z.equiv_modulo_Transitive Z.mul_mod_Proper Z.add_mod_Proper Z.modulo_equiv_modulo_Proper. + + Lemma goldilocks_mul_correct (p : Z) (p_nonzero : p <> 0) s (s_nonzero : s <> 0) (s2_modp : (s^2) mod p = (s+1) mod p) xs ys : + (eval (goldilocks_mul s xs ys)) mod p = (eval xs * eval ys) mod p. + Proof using Type*. cbv [goldilocks_mul]; Zmod_to_equiv_modulo. + repeat rewrite ?eval_mul, ?eval_add, ?eval_sub, ?eval_scmul, <-?(eval_split s xs), <-?(eval_split s ys) by assumption; ring_simplify. + setoid_rewrite s2_modp. + apply f_equal2; nsatz. Qed. +End Karatsuba. |