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modular_arithmetic.clj
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modular_arithmetic.clj
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(ns tupelo.math.modular-arithmetic
(:use tupelo.core)
(:require
[schema.core :as s]
)
(:import
[clojure.lang BigInt]))
; #todo move to i
; tupelo.math.mod.long
; tupelo.math.mod.BigInteger
; tupelo.math.mod.BigInt
;
;-----------------------------------------------------------------------------
; #todo need BigInt version?
(defn ceil-long [x] (long (Math/ceil (double x))))
(defn floor-long [x] (long (Math/floor (double x))))
(defn round-long [x] (long (Math/round (double x))))
(defn trunc-long [x] (long (.longValue (double x))))
(s/defn signum :- s/Int
"Returns either -1, 0, or +1, to indicate the sign of the input. "
[x :- s/Num]
(cond
(pos? x) +1
(neg? x) -1
:else 0))
(s/defn same-sign :- s/Bool
"Returns `true` iff x and y have the same sign, or are both zero."
[x :- s/Num
y :- s/Num]
(truthy?
(or
(and (pos? x) (pos? y))
(and (neg? x) (neg? y))
(and (zero? x) (zero? y)))))
;-----------------------------------------------------------------------------
; shortcuts for quot/mod with different return types
(s/defn mod-Long :- Long
"Computes the mod of two Long numbers, returning a Long."
[n :- Long
d :- Long] (clojure.core/mod ^Long n ^Long d))
(s/defn quot-Long :- Long
"Computes the quot of two Long numbers, returning a Long."
[n :- Long
d :- Long] (clojure.core/quot ^Long n ^Long d))
(s/defn mod-BigInteger :- BigInteger
"Computes the mod of two BigInteger numbers, returning a BigInteger."
[n :- BigInteger
d :- BigInteger] (.mod ^BigInteger n ^BigInteger d))
(s/defn quot-BigInteger :- BigInteger
"Computes the quot of two BigInteger numbers, returning a BigInteger."
[n :- BigInteger
d :- BigInteger] (.divide ^BigInteger n ^BigInteger d))
(s/defn mod-BigInt :- BigInt
"Computes the mod of two BigInt numbers, returning a BigInt."
[n :- s/Int
d :- s/Int] (mod ^BigInt (bigint n) ^BigInt (bigint d)))
(s/defn quot-BigInt :- BigInt
"Computes the quot of two BigInt numbers, returning a BigInt."
[n :- s/Int
d :- s/Int] (quot ^BigInt (bigint n) ^BigInt (bigint d)))
;-----------------------------------------------------------------------------
; shortcuts for modular add/mult with different return types
(s/defn add-mod-Long :- Long
"Adds two numbers a and b (mod N)."
[a :- Long
b :- Long
N :- Long]
(assert (and (pos? N) (< 1 N)))
(it-> (+ a b)
(mod-Long it N)))
(s/defn mult-mod-Long :- Long
"Multiply two numbers a and b (mod N)."
[a :- Long
b :- Long
N :- Long]
(assert (and (pos? N) (< 1 N)))
(it-> (* a b)
(mod-Long it N)))
(s/defn add-mod-BigInteger :- BigInteger
"Adds two numbers a and b (mod N)."
[a :- BigInteger
b :- BigInteger
N :- BigInteger]
(assert (and (pos? N) (< 1 N)))
(it-> (.add ^BigInteger a ^BigInteger b)
(mod-BigInteger it N)))
(s/defn mult-mod-BigInteger :- BigInteger
"Multiply two numbers a and b (mod N)."
[a :- BigInteger
b :- BigInteger
N :- BigInteger]
(assert (and (pos? N) (< 1 N)))
(it-> (.multiply ^BigInteger a ^BigInteger b)
(mod-BigInteger it N)))
(s/defn add-mod-BigInt :- BigInt
"Adds two numbers a and b (mod N)."
[a :- s/Int
b :- s/Int
N :- s/Int]
(assert (and (pos? N) (< 1 N)))
(it-> (+ a b)
(mod-BigInt it N)))
(s/defn mult-mod-BigInt :- BigInt
"Multiply two numbers a and b (mod N)."
[a :- s/Int
b :- s/Int
N :- s/Int]
(assert (and (pos? N) (< 1 N)))
(it-> (* a b)
(mod-BigInt it N)))
;-----------------------------------------------------------------------------
(s/defn mod-symmetric :- s/Int
"Like clojure.core/mod, but returns a result symmetric around zero [-N/2..N/2). N must be even and positive."
[i :- s/Int
N :- s/Int]
(assert (and (int? i) (int-pos? N) (even? N)))
(let [d-ovr-2 (/ N 2)
result (cond-it-> (mod i N)
(<= d-ovr-2 it) (- it N))]
result))
(s/defn mod-inverse :- s/Int
"Computes the 'inverse` y of a number x (mod N), such that `x*y (mod N)` = 1.
Uses the extended Euclid algorithm (iterative version). Assumes x and N are relatively prime. "
[x :- s/Int
N :- s/Int]
(assert (and (pos? x) (pos? N) (< x N)))
(let [N-orig N
a 1
b 0]
(if (= 1 N)
(throw (ex-info "Invalid N" (vals->map x N)))
(loop [x x
n N
a a
b b]
(if (< 1 x)
(let [x-next n
N-next (mod x n)
q (quot x n)
a-next b
b-next (- a (* q b))]
(recur x-next N-next a-next b-next))
(if (neg? a)
(+ a N-orig)
a))))))