From: kskalb@immd1.informatik.uni-erlangen.de (Klaus Kalb)
Newsgroups: comp.sources.hp48
Subject: v01i008:  ntl2 - A Number Theory Library v2.5, Part01/01
Date: 4 Aug 91 23:45:04 GMT
Followup-To: comp.sys.hp48

Submitted-by: Klaus Kalb <kskalb@immd1.informatik.uni-erlangen.de>
Posting-number: Volume 1, Issue 8
Archive-name: ntl2/part01


BEGIN_RDME NTL2.readme
---------------------------------------------------------------------------

NTL2 is a library for the pocket calculator HP48SX to perform basic 
numbertheoretic calculations. It provides modular arithmetic, factorization
of integers, chinese remaindering and some numbertheoretic functions.

---------------------------------------------------------------------------

It is provided as is, without any warranty or assertion of fitness for
any purpose. Note that this library makes use of undocumented and 
unsupported features of the HP48. This might cause data loss or even
hardware damage. Use it at your own risk.

---------------------------------------------------------------------------

This library is ShareWare. You may distribute it freely for noncommercial 
purposes only and install it into your calculator to see if it fits your
needs. You may not make any changes to the software or to the documentation,
put it into ROM, publish it in journals or sell it without written permission
of the author.

If you find this library useful and you want to keep it, send US$ 30
(or DM 40, if you prefer this) to the address shown below.  Thus
you will become a registered user and you will recieve information 
about new versions and updates to the documentation as soon
as these are available. If you do so, please include an email address
if available. If I have to deal with diskettes to send a newer version
to you, I feel free to charge an extra fee.

---------------------------------------------------------------------------

Parts of the code in this library were developed by Jurjen N. E. Bos
and are included with his kind permission. Thus this library contains
a slightly improved version of his `Ultimate Factoring Program'.  

7/24/91, -KK

---------------------------------------------------------------------------

		Klaus Kalb
		Huettendorfer Weg 14
		W-8510 Fuerth 17
		Federal Republic of Germany

		email: kskalb@informatik.uni-erlangen.de
		
---------------------------------------------------------------------------

END_RDME NTL2.readme

BEGIN_DOC NTL2.doc
;; @(#)NTL2.doc	2.5 8/1/91



 This is the complete documentation of the library NTL2 version 2.5.
 The ID of this libarary is 1672 and the title reads `NTL2 2.5 Number Theory`

 The command `AboutNTL2` should display the following screen:

 		NTL2 2.5 Number Theory
		created 08/01/91 22:59
		(C) 1991 by Klaus Kalb
		Credits to Jurjen Bos
		* This is ShareWare *

 and return the version identification string "2.5" to level 1.



      ***********************************************************
      *                                                         *
      *   This software is ShareWare. Read the README-File for  *
      *   details and disclaimers.                              *
      *                                                         *
      *   This library attaches itself automatically to the     *
      *   'HOME'-directory. Consult your HP48 Owner's Manual    *
      *   for details about libraries.                          *
      *                                                         *
      ***********************************************************


=========================================================================

	AboutNTL2
		Displays a screen full of information about NTL2 and
		creates a temporary menu to access the various pages
		of the NTL2 menu.  
		The current version ID string "2.5" is returned.
		If an entry in the temporary menu is executed with
		the string "2.5" in level 1, this string is dropped
		and some info about registration is displayed.

		


							(Global.star 2.3)
=========================================================================
==                                                                     ==
==                     Setting and Changing the Modul                  ==
==                                                                     ==
=========================================================================

 Contents of 'Data.NTL2' :
	1 : Modul as entered			
	2 : Modul as binary integer		
	3 : Phi(Modul)				
	4 : Lambda(Modul)			
	5 : Factorization of Modul		
	6 : Factorization of Lambda(Modul)	
	7 : Splitting for CRA			
	8 : Data for CRA			
	9 : Splitting in Binary			

=========================================================================

	MSet
		Sets the modulus to the argument. All entries in Data.NTL2
		will be destroyed.
		If the argument is a list, the entries must be relatively
		prime. In this case the modulus will be set to the product
		of all entries in the list (if this number is smaller 
		then 2^63) and the chinese remainder support will be
		initialized to the given factorization.

		The modulus determines the representation of residues:
		---	If it is positive, then the least positive residues
			will be used.
		---	If it is negative, then the residue with the least
			absolute value will be used.
		---	If its a binary, all residues will be represented
			as binaries.

		Note that even if the modulus has been set to a real number,
		all arithmetic will be performed with binaries. So be sure
		that the current wordsize is large enough.

=========================================================================

	MGet
		Retrieves the current modulus.

=========================================================================

	MTgl
		If the current modulus is a binary number less or equal to
		10^12, the modulus will be switched to its negative real
		value.
		If the current modulus is a real number, its sign will be
		changed.
		This function allows changing the representation without
		destroying the values stored in Data.NTL2.

=========================================================================

	MPrg
		Purges Data.NTL2 from the current directory.

=========================================================================


							(MArith1.star 2.4)
=========================================================================
==                                                                     ==
==                    Basic Modular Arithmetik                         ==
==                                                                     ==
=========================================================================

	\->Mod

		Converts its argument in the current modular representation
		Algebraics containing only global and local names and the
		basic functions +,-,*,/,INV() and NEG() will be evaluated.
		Any not integral reals will be converted using \->Q.
		Note that you have to be very careful about roundoffs 
		when you apply \->MOD to a non integral real. 

		If the argument is a list, \->MOD will be applied to
		each element of the list and the individual results
		will be combined to a new list.

		If the argument is a real array or a vector, \->MOD 
		will be applied to any entry. 

		mode maps: 
			#m			-> r (binary or real)
			m			-> r (binary or real)
			`Formula`		-> r (binary or real)
			{ o1 o2 ... }		-> { r1 r2 ... }
			[ [ real matrix ] ]	-> [ [ real matrix ] ] 

=========================================================================

	MAdd

		Modular addition.
		Supports chinese remainder representation.

		mode maps: 
			#a #b			-> a+b (binary or real)
			a b			-> a+b (binary or real)
			#a b			-> a+b (binary or real)
			#a #b			-> a+b (binary or real)
			{ a_1 ... a_n } b	-> { a+b_1 ... a+b_n }
			{ a_1 ... a_n } #b	-> { a+b_1 ... a+b_n }
			#a { b_1 ... b_n } 	-> { a+b_1 ... a+b_n }
			a { b_1 ... b_n } 	-> { a+b_1 ... a+b_n }
			{ a_1 ... } { b_1 ... } -> { a+b_1 ... a+b_n }

=========================================================================

	MSub

		Modular subtraction.
		Supports chinese remainder representation.

		mode maps: 
			#a #b			-> a-b (binary or real)
			a b			-> a-b (binary or real)
			#a b			-> a-b (binary or real)
			#a #b			-> a-b (binary or real)
			{ a_1 ... a_n } b	-> { a-b_1 ... a-b_n }
			{ a_1 ... a_n } #b	-> { a-b_1 ... a-b_n }
			#a { b_1 ... b_n } 	-> { a-b_1 ... a-b_n }
			a { b_1 ... b_n } 	-> { a-b_1 ... a-b_n }
			{ a_1 ... } { b_1 ... } -> { a-b_1 ... a-b_n }

=====================================================================

	MMul

		Modular multiplication.
		Supports chinese remainder representation.

		mode maps: 
			#a #b			-> a*b (binary or real)
			a b			-> a*b (binary or real)
			#a b			-> a*b (binary or real)
			#a #b			-> a*b (binary or real)
			{ a_1 ... a_n } b	-> { a*b_1 ... a*b_n }
			{ a_1 ... a_n } #b	-> { a*b_1 ... a*b_n }
			#a { b_1 ... b_n } 	-> { a*b_1 ... a*b_n }
			a { b_1 ... b_n } 	-> { a*b_1 ... a*b_n }
			{ a_1 ... } { b_1 ... } -> { a*b_1 ... a*b_n }

=========================================================================

	MDiv

		Modular division.
		Note that the divisor must be a unit. 
		Supports chinese remainder representation.

		mode maps: 
			#a #b			-> a/b (binary or real)
			a b			-> a/b (binary or real)
			#a b			-> a/b (binary or real)
			#a #b			-> a/b (binary or real)
			{ a_1 ... a_n } b	-> { a/b_1 ... a/b_n }
			{ a_1 ... a_n } #b	-> { a/b_1 ... a/b_n }
			#a { b_1 ... b_n } 	-> { a/b_1 ... a/b_n }
			a { b_1 ... b_n } 	-> { a/b_1 ... a/b_n }
			{ a_1 ... } { b_1 ... } -> { a/b_1 ... a/b_n }

=========================================================================

	MExp

		Modular exponentation.
		Accepts negative exponents as well as positive ones.
		Supports chinese remainder representation for the base only.

		mode maps: 
			#b #e			-> b^e (binary or real)
			b e			-> b^e (binary or real)
			#b e			-> b^e (binary or real)
			b #e			-> b^e (binary or real)
			{ b_1 ... b_n }	#e	-> { b^e_1 ... b^e_n }
			{ b_1 ... b_n }	e	-> { b^e_1 ... b^e_n }

=========================================================================

	MInv

		Modular inversion.
		Calculates the multplicative inverse of the argument using
		an extended GCD algorithm.
		Supports chinese remainder representation.

		mode maps: 
			#m			-> r (binary or real)
			m			-> r (binary or real)
			{ m_1 m_2 ... m_n }	-> { r_1 r_2 ... r_n }

=========================================================================

	MNeg

		Multiplies its argument with -1.
		Supports chinese remainder representation.

		mode maps: 
			#m			-> r (binary or real)
			m			-> r (binary or real)
			{ m_1 m_2 ... m_n }	-> { r_1 r_2 ... r_n }

=========================================================================


							(MArith2.star 2.1)
=========================================================================
==                                                                     ==
==                   Advanced Modular Arithmetic                       ==
==                                                                     ==
=========================================================================

	M\v/

		Extracts the square root of its argument (if possible)
		Needs an odd modulus.
		This needs the 'standard' chinese remainder setup.
	
		mode maps:
			#x			-> r (binary or real)
			x			-> r (binary or real)

=========================================================================

	MSq?

		Checks wether its argument is a square modulo the current
		modulus. Needs an odd modulus.
	
		mode maps:
			#x			-> 0/1
			x			-> 0/1

=========================================================================

	MOrd

		Calculates the order of the given element.  
	
		mode maps:
			#x			-> p (binary or real)
			x			-> p (binary or real)

=========================================================================

	MNPr

		Returns the next primitive element modulo the current  
		modulus. If there are no primitive elements, an error
		will be generated.
	
		mode maps:
			#x			-> p (binary or real)
			x			-> p (binary or real)

=========================================================================

	MPr?

		Returns 1 if its argument is a primitive element modulo 
		the current modulus. 
	
		mode maps:
			#x			-> 0/1
			x			-> 0/1

=========================================================================


							(Chinese.star 2.1)
=========================================================================
==                                                                     ==
==                    Chinese Remainder Techniques                     ==
==                                                                     ==
=========================================================================

	CSet

		Sets the chinese remainder mode.
		The argument is a list of pairwise relatively prime integers.
		Note that binaries and integral reals (positive and negative) 
		are allowed in the input list.
		If a modul is set, the product of all numbers in the list
		must match the modul. If not, the modul is set to this 
		product as a binary integer if this product can be
		represented in 63 bits.

		mode maps:
			{ r_1 ... }			-> 

=========================================================================

	CGet

		recalls the current chinese remainder setting to the stack

		mode maps:
						-> { r_1 ... }

=========================================================================

	\->Crm

		Converts a number into the current chinese remainder
		representation. The result will be a list of residues
		according to the call to CSet. If CSet hasn't been called,
		the current modul will be factored and a default setting
		will be made up.

		mode maps:
			m			-> { r_1 ... }
			#m			-> { r_1 ... }

=========================================================================

	Crm\->

		Converts from the chineses remainder representation back
		to the normal modular representation.
		This is done by the well known chinese remainder algorithm.
		The input must be a list of real or binary integers

		mode maps:
			{ r_1 ... } 		-> m (binary or real)

=========================================================================


							(Numtheo.star 2.1)
=========================================================================
==                                                                     ==
==                    Basic Numbertheoretic Routines                   ==
==                                                                     ==
=========================================================================

	BGcd

		Calculates the greatest common divisor of two numbers.
		This is done by a binary algorithm programmed in ML
		
		mode maps:
			#a #b			-> #d
			a b			-> d
			#a b			-> #d
			a #b			-> #d

=========================================================================

	XGcd

		Calculates the greatest common divisor of two numbers
		and gives coefficients for a representation of the gcd
		as linear combination of the arguments. 
			gcd(a,b) = d = t*b-s*a with 0<=t<=a and 0<=s<b
		Both arguments must be positive.
		
		mode maps:
			#a #b			-> #s #t #d
			a b			-> s t d

=========================================================================

	Lcm

		Calculates the least common multiple of two numbers.
		The result will always be positive. 
		BUGS: an overflow isn't detected

		mode maps:
			#a #b			-> #r
			a #b			-> #r
			#a b			-> #r
			a b			-> r

=========================================================================


							(NumFnc.star 2.1)
=========================================================================
==                                                                     ==
==                 Number Theoeretic Functions                         ==
==                                                                     ==
=========================================================================

	Ntf\Gh	

		Calculates the Euler totient function.
		If the argument is a list, the entries must be in ascending
		order.

		mode maps: 
			#m			-> #Phi(m)
			m			-> Phi(m)
			{ p_1 ... p_n }		-> Phi(p_1 * ... * p_n)

=========================================================================

	Ntf\Gl	

		Calculates the Charmichal's Lambda function
		The Charmicheal Lambda function gives the exponent of
		the group of units modulo m.
		If the argument is a list, the entries must be in ascending
		order.

		mode maps: 
			#m			-> #Lambda(m)
			m			-> Lambda(m)
			{ p_1 ... p_n }		-> Lambda(p_1 * ... * p_n)

=========================================================================

	Ntf\Gs	

		Calculates the number of divisors of the argument.
		If the argument is a list, the entries must be in ascending
		order.

		mode maps: 
			#m			-> #Sigma(m)
			m			-> Sigma(m)
			{ p_1 ... p_n }		-> Sigma(p_1 * ... * p_n)

=========================================================================

	Ntf\Gm	

		Calculates the Moebius function
		If the argument is a list, the entries must be in ascending
		order.

		mode maps: 
			#m			-> #Mu(m)
			m			-> Mu(m)
			{ p_1 ... p_n }		-> Mu(p_1 * ... * p_n)

=========================================================================


                                                    (Jacobi.star 2.2)
=========================================================================

    Jacobi

		Calculates the Jacobi symbol for its arguments using
		the quadratic reciprocity law. Note that the value
		is zero if the arguments aren't relatively prime
		and that the value in level 1 must be an odd number.
		The result will always equal +1, -1 or 0.
 		
            mode maps:
                    a b                     -> (a/b) 
                    #a #b                   -> (a/b) 
                    a #b                    -> (a/b) 
                    #a b                     -> (a/b) 



							(Prime.star 2.1)
=========================================================================
==                                                                     ==
==                 Primality Testing and Factoring                     ==
==                                                                     ==
=========================================================================

	Prm?

		Tests wether its argument is a prime number
		Returns 1 if it's a prime, else 0
		If the number to test is greater then 25*10^9, the
		test may take up to several minutes.
		Negative primes are recognized correctly

		mode maps: 
			#m			-> 0/1
			m			-> 0/1

=========================================================================

	NPrm

		Finds the smallest prime greater then the absolute value
		of the argument

		mode maps: 
			#x			-> #p
			x			-> p

=========================================================================

	PPrm

		Finds the greatest prime less then the absolute value
		of the argument

		mode maps: 
			#x			-> #p
			x			-> p

=========================================================================

	Factor

		Factors its argument into primes
		The output will be a list containing all prime divisors
		(repeated according to their multiplicity) in ascending
		order.
		The entries in the list will have the same type as the
		argument.
		The number to be factored must be positive.

		mode maps: 
			#x			-> { #p_1 #p_2 ... }
			x			-> { p_1 p_2 ... }

=========================================================================


						    (MRand.star 2.2)
=========================================================================
==                                                                     ==
==                Additional Modular Arithmetic                        ==
==                                                                     ==
=========================================================================

    MRand

		Generates a random number modulo the current module.

	 	mode maps:
					    -> r (binary or real)

=========================================================================

END_DOC

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5277640040D8D4F64650040D414464660040D435572670040D4D457C680040D4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"
END_ASC
------------------------------------------------------------------------------
   Klaus Kalb    | mail :  IMMD1 / Martenstr. 3 / W-8520 Erlangen / Germany   
                 | email:  kskalb@immd1.informatik.uni-erlangen.de   
------------------------------------------------------------------------------

exit 0 # Just in case...