Document miller-rabin, more unit tests for some corner cases
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basis/math/miller-rabin
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! Copyright (C) 2009 Doug Coleman.
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! See http://factorcode.org/license.txt for BSD license.
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USING: help.markup help.syntax kernel sequences math ;
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IN: math.miller-rabin
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HELP: find-relative-prime
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{ $values
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{ "n" integer }
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{ "p" integer }
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}
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{ $description "Returns a number that is relatively prime to " { $snippet "n" } "." } ;
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HELP: find-relative-prime*
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{ $values
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{ "n" integer } { "guess" integer }
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{ "p" integer }
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}
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{ $description "Returns a number that is relatively prime to " { $snippet "n" } ", starting by trying " { $snippet "guess" } "." } ;
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HELP: miller-rabin
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{ $values
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{ "n" integer }
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{ "?" "a boolean" }
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}
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{ $description "Returns true if the number is a prime. Calls " { $link miller-rabin* } " with a default of 10 Miller-Rabin tests." } ;
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{ miller-rabin miller-rabin* } related-words
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HELP: miller-rabin*
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{ $values
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{ "n" integer } { "numtrials" integer }
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{ "?" "a boolean" }
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}
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{ $description "Performs " { $snippet "numtrials" } " trials of the Miller-Rabin probabilistic primality test algorithm and returns true if prime." } ;
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HELP: next-prime
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{ $values
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{ "n" integer }
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{ "p" integer }
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}
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{ $description "Tests consecutive numbers for primality with " { $link miller-rabin } " and returns the next prime." } ;
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HELP: next-safe-prime
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{ $values
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{ "n" integer }
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{ "q" integer }
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}
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{ $description "Tests consecutive numbers and returns the next safe prime. A safe prime is desirable in cryptography applications such as Diffie-Hellman and SRP6." } ;
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HELP: random-bits*
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{ $values
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{ "numbits" integer }
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{ "n" integer }
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}
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{ $description "Returns an integer exactly " { $snippet "numbits" } " in length, with the topmost bit set to one." } ;
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HELP: random-prime
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{ $values
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{ "numbits" integer }
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{ "p" integer }
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}
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{ $description "Returns a prime number exactly " { $snippet "numbits" } " bits in length, with the topmost bit set to one." } ;
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HELP: random-safe-prime
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{ $values
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{ "numbits" integer }
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{ "p" integer }
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}
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{ $description "Returns a safe prime number " { $snippet "numbits" } " bits in length, with the topmost bit set to one." } ;
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HELP: safe-prime?
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{ $values
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{ "q" integer }
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{ "?" "a boolean" }
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}
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{ $description "Tests whether the number is a safe prime. A safe prime " { $snippet "p" } " must be prime, as must " { $snippet "(p - 1) / 2" } "." } ;
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HELP: unique-primes
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{ $values
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{ "numbits" integer } { "n" integer }
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{ "seq" sequence }
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}
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{ $description "Generates a sequence of " { $snippet "n" } " unique prime numbers with exactly " { $snippet "numbits" } " bits." } ;
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ARTICLE: "math.miller-rabin" "Miller-Rabin probabilistic primality test"
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"The " { $vocab-link "math.miller-rabin" } " vocabulary implements the Miller-Rabin probabilistic primality test and utility words that use it in order to generate random prime numbers." $nl
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"The Miller-Rabin probabilistic primality test:"
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{ $subsection miller-rabin }
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{ $subsection miller-rabin* }
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"Generating relative prime numbers:"
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{ $subsection find-relative-prime }
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{ $subsection find-relative-prime* }
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"Generating prime numbers:"
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{ $subsection next-prime }
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{ $subsection random-prime }
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"Generating safe prime numbers:"
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{ $subsection next-safe-prime }
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{ $subsection random-safe-prime } ;
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ABOUT: "math.miller-rabin"
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@ -1,4 +1,5 @@
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USING: math.miller-rabin tools.test kernel sequences ;
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USING: math.miller-rabin tools.test kernel sequences
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math.miller-rabin.private math ;
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IN: math.miller-rabin.tests
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[ f ] [ 473155932665450549999756893736999469773678960651272093993257221235459777950185377130233556540099119926369437865330559863 miller-rabin ] unit-test
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@ -6,6 +7,9 @@ IN: math.miller-rabin.tests
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[ t ] [ 3 miller-rabin ] unit-test
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[ f ] [ 36 miller-rabin ] unit-test
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[ t ] [ 37 miller-rabin ] unit-test
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[ 2 ] [ 1 next-prime ] unit-test
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[ 3 ] [ 2 next-prime ] unit-test
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[ 5 ] [ 3 next-prime ] unit-test
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[ 101 ] [ 100 next-prime ] unit-test
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[ t ] [ 2135623355842621559 miller-rabin ] unit-test
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[ 100000000000031 ] [ 100000000000000 next-prime ] unit-test
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@ -14,6 +18,12 @@ IN: math.miller-rabin.tests
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[ f ] [ 862 safe-prime? ] unit-test
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[ t ] [ 7 safe-prime? ] unit-test
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[ f ] [ 31 safe-prime? ] unit-test
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[ t ] [ 47 safe-prime-candidate? ] unit-test
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[ t ] [ 47 safe-prime? ] unit-test
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[ t ] [ 863 safe-prime? ] unit-test
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[ f ] [ 1000 [ drop 15 miller-rabin ] any? ] unit-test
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[ 47 ] [ 31 next-safe-prime ] unit-test
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[ 49 ] [ 50 random-prime log2 ] unit-test
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[ 49 ] [ 50 random-bits* log2 ] unit-test
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@ -1,15 +1,20 @@
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! Copyright (C) 2008 Doug Coleman.
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! See http://factorcode.org/license.txt for BSD license.
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USING: combinators kernel locals math math.functions math.ranges
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random sequences sets combinators.short-circuit math.bitwise ;
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random sequences sets combinators.short-circuit math.bitwise
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math math.order ;
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IN: math.miller-rabin
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<PRIVATE
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: >odd ( n -- int ) dup even? [ 1 + ] when ; foldable
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: >odd ( n -- int ) 0 set-bit ; foldable
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: >even ( n -- int ) 0 clear-bit ; foldable
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: next-even ( m -- n ) >even 2 + ;
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: next-odd ( m -- n ) dup even? [ 1 + ] [ 2 + ] if ;
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TUPLE: positive-even-expected n ;
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:: (miller-rabin) ( n trials -- ? )
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@ -18,7 +23,7 @@ TUPLE: positive-even-expected n ;
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0 :> a!
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trials [
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drop
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n 1 - [1,b] random a!
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2 n 2 - [a,b] random a!
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a s n ^mod 1 = [
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f
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] [
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@ -30,8 +35,6 @@ TUPLE: positive-even-expected n ;
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PRIVATE>
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: next-odd ( m -- n ) dup even? [ 1 + ] [ 2 + ] if ;
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: miller-rabin* ( n numtrials -- ? )
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over {
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{ [ dup 1 <= ] [ 3drop f ] }
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@ -42,11 +45,21 @@ PRIVATE>
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: miller-rabin ( n -- ? ) 10 miller-rabin* ;
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ERROR: prime-range-error n ;
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: next-prime ( n -- p )
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next-odd dup miller-rabin [ next-prime ] unless ;
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dup 1 < [ prime-range-error ] when
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dup 1 = [
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drop 2
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] [
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next-odd dup miller-rabin [ next-prime ] unless
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] if ;
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: random-bits* ( numbits -- n )
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1 - [ random-bits ] keep set-bit ;
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: random-prime ( numbits -- p )
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random-bits next-prime ;
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random-bits* next-prime ;
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ERROR: no-relative-prime n ;
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@ -80,10 +93,7 @@ ERROR: too-few-primes ;
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<PRIVATE
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: >safe-prime-form ( q -- p ) 2 * 1 + ;
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: safe-prime-candidate? ( n -- ? )
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>safe-prime-form
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1 + 6 divisor? ;
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: next-safe-prime-candidate ( n -- candidate )
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@ -99,14 +109,8 @@ PRIVATE>
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} 1&& ;
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: next-safe-prime ( n -- q )
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1 - >even 2 /
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next-safe-prime-candidate
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dup >safe-prime-form
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dup miller-rabin
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[ nip ] [ drop next-safe-prime ] if ;
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: random-bits* ( numbits -- n )
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[ random-bits ] keep set-bit ;
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dup safe-prime? [ next-safe-prime ] unless ;
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: random-safe-prime ( numbits -- p )
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1- random-bits* next-safe-prime ;
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random-bits* next-safe-prime ;
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