f2dc2027f6
in the original GitHub fork: https://github.com/ntoronto/racket Some things about this are known to be broken (most egregious is that the array tests DO NOT RUN because of a problem in typed/rackunit), about half has no coverage in the tests, and half has no documentation. Fixes and docs are coming. This is committed now to allow others to find errors and inconsistency in the things that appear to be working, and to give the author a (rather incomplete) sense of closure.
179 lines
6.8 KiB
Racket
179 lines
6.8 KiB
Racket
#lang racket/base
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(provide with-modulus
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current-modulus
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modular-inverse
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modular-expt
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inline-mod+
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inline-mod*
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inline-mod-
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inline-mod/
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inline-modsqr
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inline-modexpt
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inline-mod
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inline-mod=
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inline-mod<
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inline-mod<=
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inline-mod>
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inline-mod>=)
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(module typed-defs typed/racket/base
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(require racket/performance-hint
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"divisibility.rkt")
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(provide (all-defined-out))
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(: current-modulus-param (Parameterof Positive-Integer))
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(define current-modulus-param (make-parameter 1))
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(: current-modulus (-> Positive-Integer))
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(begin-encourage-inline
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(define (current-modulus) (current-modulus-param)))
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; THEOREM
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; If gcd(a,n)=1 then there exist b such that
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; ab=1 mod n
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; The number b is called an inverse of a modulo n.
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(: modular-inverse* (Positive-Integer Integer -> Natural))
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(define (modular-inverse* n a)
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(cond [(zero? a) (raise-argument-error 'modular-inverse "nonzero Integer" 0 a n)]
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[(coprime? n a) (modulo (car (bezout a n)) n)]
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[else (error 'modular-inverse "expected argument that is coprime to modulus ~e; given ~e"
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n a)]))
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(: modular-expt* (Positive-Integer Integer Integer -> Natural))
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;; Exponentiate by repeated modular multiplication and squaring
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(define (modular-expt* n a b)
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(cond [(b . < . 0) (raise-argument-error 'modular-expt "Natural" 1 a b n)]
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[else
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(let loop ([a a] [b b])
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(cond [(b . <= . 1) (if (zero? b) (modulo 1 n) (modulo a n))]
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[(even? b) (define c (loop a (quotient b 2)))
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(modulo (* c c) n)]
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[else (modulo (* a (loop a (sub1 b))) n)]))]))
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(: modular-const* (Positive-Integer Exact-Rational -> Natural))
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(define (modular-const* n a)
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(cond [(integer? a) (modulo a n)]
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[else (modulo (* (numerator a) (modular-inverse* n (denominator a))) n)]))
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(: modular-inverse (Integer Integer -> Natural))
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;; Return b, where a*b=1 mod n and b in {0,...,n-1}
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(define (modular-inverse a n)
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(cond [(n . <= . 0) (raise-argument-error 'modular-inverse "Positive-Integer" 1 a n)]
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[else (modular-inverse* n a)]))
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(: modular-expt (Integer Integer Integer -> Natural))
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(define (modular-expt a b n)
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(cond [(n . <= . 0) (raise-argument-error 'modular-expt "Positive-Integer" 2 a b n)]
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[else (modular-expt* n a b)]))
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)
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(module untyped-defs racket/base
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(require (for-syntax racket/base)
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racket/stxparam
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(submod ".." typed-defs))
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(provide (all-defined-out))
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(define-syntax-parameter current-modulus-id #f)
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;; Sets the `current-modulus-param' and `current-modulus-id'
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(define-syntax (with-modulus stx)
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(syntax-case stx ()
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[(_ modulus . body)
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(syntax/loc stx
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(let ([n modulus])
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(syntax-parameterize ([current-modulus-id #'n])
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(parameterize ([current-modulus-param n])
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. body))))]))
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;; Checks for `current-modulus-id'; if an identifier, uses the identifier's bound value for the
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;; modulus; otherwise, gets the current modulus and sets `current-modulus-id' for inner expressions
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(define-syntax (inline-mod-op stx)
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(syntax-case stx ()
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[(_ op-macro a ...)
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(with-syntax ([n (syntax-parameter-value #'current-modulus-id)])
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(cond [(identifier? #'n) (syntax/loc stx (op-macro n a ...))]
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[else
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(syntax/loc stx
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(let ([m (current-modulus-param)])
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(syntax-parameterize ([current-modulus-id #'m])
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(op-macro m a ...))))]))]))
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(define-syntax (fold-mod-op stx)
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(syntax-case stx ()
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[(_ op n a b)
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(syntax/loc stx (modulo (op a b) n))]
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[(_ op n a b cs ...)
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(syntax/loc stx
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(fold-mod-op op n (modulo (op a b) n) cs ...))]))
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(define-syntax (modular-compare stx)
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(syntax-case stx ()
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[(_ op n a) (syntax/loc stx #t)]
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[(_ op n a b) (syntax/loc stx (op (modulo a n) (modulo b n)))]
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[(_ op n a b-expr bs ...)
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(syntax/loc stx
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(let ([b (modulo b-expr n)])
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(and (op (modulo a n) b)
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(fold-mod-compare-op op n b bs ...))))]))
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(define-syntax (fold-mod-compare-op stx)
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(syntax-case stx ()
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[(_ op n a b)
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(syntax/loc stx (op a (modulo b n)))]
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[(_ op n a b-expr bs ...)
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(syntax/loc stx
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(let ([b (modulo b-expr n)])
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(and (op a b)
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(fold-mod-compare-op op n b bs ...))))]))
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(define-syntax (modular+ stx)
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(syntax-case stx ()
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[(_ n) (syntax/loc stx 0)]
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[(_ n a) (syntax/loc stx (modulo a n))]
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[(_ n a ...) (syntax/loc stx (fold-mod-op + n a ...))]))
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(define-syntax (modular* stx)
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(syntax-case stx ()
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[(_ n) (syntax/loc stx 1)]
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[(_ n a) (syntax/loc stx (modulo a n))]
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[(_ n a ...) (syntax/loc stx (fold-mod-op * n a ...))]))
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(define-syntax (modular- stx)
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(syntax-case stx ()
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[(_ n a) (syntax/loc stx (modulo (- a) n))]
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[(_ n a b ...) (syntax/loc stx (fold-mod-op - n a b ...))]))
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(define-syntax (modular/ stx)
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(syntax-case stx ()
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[(_ n a) (syntax/loc stx (modular-inverse n a))]
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[(_ n a b ...) (syntax/loc stx
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(modular* n a (modular-inverse* n (modular* n b ...))))]))
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(define-syntax-rule (modular-sqr n a) (modulo (* a a) n))
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(define-syntax-rule (modular= n a b ...) (modular-compare = n a b ...))
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(define-syntax-rule (modular< n a b ...) (modular-compare < n a b ...))
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(define-syntax-rule (modular<= n a b ...) (modular-compare <= n a b ...))
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(define-syntax-rule (modular> n a b ...) (modular-compare > n a b ...))
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(define-syntax-rule (modular>= n a b ...) (modular-compare <= n a b ...))
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(define-syntax-rule (inline-mod+ a ...) (inline-mod-op modular+ a ...))
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(define-syntax-rule (inline-mod* a ...) (inline-mod-op modular* a ...))
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(define-syntax-rule (inline-mod- a b ...) (inline-mod-op modular- a b ...))
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(define-syntax-rule (inline-mod/ a b ...) (inline-mod-op modular/ a b ...))
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(define-syntax-rule (inline-modsqr a) (inline-mod-op modular-sqr a))
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(define-syntax-rule (inline-modexpt a b) (inline-mod-op modular-expt* a b))
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(define-syntax-rule (inline-mod a) (inline-mod-op modular-const* a))
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(define-syntax-rule (inline-mod= a b ...) (inline-mod-op modular= a b ...))
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(define-syntax-rule (inline-mod< a b ...) (inline-mod-op modular< a b ...))
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(define-syntax-rule (inline-mod<= a b ...) (inline-mod-op modular<= a b ...))
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(define-syntax-rule (inline-mod> a b ...) (inline-mod-op modular> a b ...))
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(define-syntax-rule (inline-mod>= a b ...) (inline-mod-op modular>= a b ...))
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)
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(require (submod "." typed-defs)
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(submod "." untyped-defs))
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