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NUMBERS MODULE version 0.08 =========================== (C) Copyright Nick Craig-Wood 1992 The module "Numbers" and this documentation "NumModTxt" are public domain. You may distribute them freely provided that 1) both files are always kept together 2) the files are not altered in any way 3) no profit is made If you wish to break any of these conditions get in touch with me at the address below first. The software and documentation was first published in BBC Acorn User Magazine, September 1992, and for more information see that issue. These routines started life in 1989 written in C on the university mainframe in response to a "Numbers Count" puzzle in PCW. The problem (or part of it) was to calculate the biggest prime you could find of the form N!+1. However the routines did not get finished in time for the puzzle deadline. In the quest for speed I converted the routines to 68000 code for my Atari ST, and then a year later into ARM code. With an ARM3 the routines run at the same speed as they did on the mainframe! Numbers are held in application memory area in a heap, and maintained by OS_Heap. Before the module can be used, it must be told about this heap with Num_InitHeap. Numbers come in two pieces, the head and the tail. All numbers have a head, but need not have a tail. The head of the number is a short block in the heap, which points to the tail and keeps the length and sign of the number, and various other things. Numbers are passed by reference only, as pointers to the head of the number. These are refered to as NUMs. For example NUM r0 indicates that r0 is a pointer to the head of a number. Since NUMs are passed by reference only, if you pass them into procedures and then alter the values of the input parameters, you are actually altering the NUM. So to avoid problems, if you write a procedure which takes NUMs as parameters for both input and output, it is best to define some temporary NUMs using Num_Init for the output, and then at the end Num_Swap the results into the correct place, and Num_Remove the temporary variables. All the internal routines of the module work like this, so there is never any problem with passing the same NUM as input and output. Some of the routines take either NUMs or scalars as arguments. Scalars are normal signed or unsigned 32 bit numbers. All routines expect NUMs to be tidy (except Num_Tidy). (A num is tidy if its most significant digit is non-zero, and if it is zero, then it is positive, and of length 1. Num_Tidy accomplishes this. It is unlikely that a user will need this function.) Any error from the numbers module will be prefixed with "Numbers: ". If you pass an invalid NUM to any of the routines, the chances are that OS_Heap will reply with an error, but not one that makes sense always. The most confusing is "Numbers: Heap Full" so beware! See the end for a short example program. If you find any bugs, have any comments or queries or wish to use the module in commercial software please write to Nick Craig-Wood 4 Victoria Street Wolverton Bucks MK12 5HL. ----------------------------------------------------------------------------- NUMBERS SWI =========== On Entry: r0-r3 are parameters On Exit: r0-r3 are returned as results or corrupted r4-r14 are preserved Interrupts: Interrupts are enabled Fast interrupts are enabled Processor mode: Processor is in SVC mode Re-entrancy: SWIs are not re-entrant Use: See QUICK REFERENCE, and DEFINITIONS for details All SWIs that are capable of looping respond to ESCAPE See ERRORS section for details of errors returned ----------------------------------------------------------------------------- QUICK REFERENCE =============== NUMBERS HEAP: area in application workspace where all the number are kept HEAD: 16 byte parameter block in the numbers heap pointing to TAIL: variable length block containing the value of the number NUM: meaning a 32 bit integer pointing to a HEAD INT: meaning a 32 bit integer SCALAR: meaning a 32 bit integer FLAG: is either 0 for false, or 1 for true STRING: a pointer to a 0 terminated ASCII string a,b,c,d represent pointers to NUMs i,j,k represent SCALARS p,q represent a pointer to memory f represents a FLAG h represents a HeapPointer as returned by Num_HeapInit in r0 SWI Parameters Results Comments === ========== ======= ======== Num_Author - - Prints out some info Num_InitHeap p,i h,a,b,c h<-HeapPointer, a<-0, b<-1, c<-2 Num_MakeSmallPrimes i j Makes primes up to i, # found to j (All the SWIs below need a valid NUM or HeapPointer in r0) Num_Allocate a,i - Allocates i words to TAIL of a Num_Deallocate a - Removes the TAIL of a Num_Set a,i - Sets a to signed i Num_USet a,i - Sets a to unsigned i Num_Init h a Makes a HEAD and TAIL and pointer a Num_Remove a - Removes the HEAD and TAIL of a Num_Equals a,i f Returns truth of a = i Num_Swap a,b - Swaps a and b. This is quick. Num_Move a,b - Sets b to a. Not so quick. Num_Clear a - Sets TAIL of a to all zeroes Num_Tidy a - Reduces a to shortest length Num_UCmp a,b i Unsigned compare of a-b to i Num_Cmp a,b i Returns the sign of a-b to i Num_ScalarCmp a,i j Returns the sign of a-i to j Num_ScalarAdd a,i,c - c <- a + i Num_ScalarSub a,i,c - c <- a - i Num_ScalarMul a,i,c - c <- a * i Num_ScalarDiv a,i,c j c <- a / i, j <- a MOD i Num_ScalarMod a,i j j <- a MOD i Num_SmallFactorN a,i k k=smallest factor of a or 0, try i Num_SmallFactor a k k=smallest factor of a or 0, try all Num_Add a,b,c - c <- a + b Num_Sub a,b,c - c <- a - b Num_Mul a,b,c - c <- a * b Num_Div a,b,c,d - c <- a / b, d <- a MOD b Num_Mod a,b,c - c <- a MOD b Num_Dump a - Prints a in hex, and other info Num_ToString a p,q Makes STRING of a to p, end=q After use do SYS "Num_Deallocate",b Num_Print a - Prints a in decimal Num_FromString a,p f Converts STRING at p to a, f=ok Num_Input a f Gets a decimal input to a, f=ok Num_Info h - Prints memory usage info Num_RndScalar h i Makes a random number 0-&FFFFFFFF Num_SetSeed h,i - Sets seed of rnd generator Num_Rnd a,b - Makes random number to a, 0 <= a < b Num_Gcd a,b,c - c <- greatest common divisor a,b Num_Sqr a,b - b <- square root of a Num_Pow a,b,c - c <- a ^ b (a to the power b) Num_PowMod a,d,c,b - d <- (a^b) MOD c Num_Factorial a,b - b <- a! = a*(a-1)*(a-1)*..*3*2*1 Num_Inv a,b,c,d - Finds c,d: a*c MOD b=d & d=GCD(a,b) Num_FermatTest a,i j Computes truth of i^(a-1) MOD a = 1 Num_ProbablyPrime a j j <- 0 if not prime, 1 probablyprime ----------------------------------------------------------------------------- DEFINITIONS =========== Num_InitHeap ------------ This expects r0 as a pointer to a block of memory to be used as a heap, and r1 as the length of this memory in bytes. This returns a pointer to the heap in r0 and some nums which are useful consants r1 <- zero, r2 <- one, r3 <- two. The HeapPointer returned in r0 is required by some of the other routines. Num_Allocate ------------ This updates NUM r0 to point to a new area of memory of length r1 (in 32-bit words) Num_Deallocate -------------- This releases the memory used by a NUM (tail), and sets its pointers to NULL Num_Set ------- This sets the NUM supplied in r0 to the signed scalar supplied in r1 Num_Uset -------- This sets the NUM supplied in r0 to the unsigned scalar supplied in r1 Num_Init -------- Expects r0=HeapPointer This returns the address of a new NUM in r0 For creating new or LOCAL variables Num_Remove ---------- This removes the allocation for NUM r0 and removes NUM r0 (For removing LOCAL variables) Num_Equals ---------- This function returns TRUE in r0 if NUM r0 is equal to the single element supplied in r1 Num_Swap -------- This swaps the two NUMs in r0 and r1. It is a fast way of getting infromation out of temporary variables, before destroying them Num_Move -------- This moves NUM r0 to NUM r1. It does this by actually copying the NUM. Num_Clear --------- This clears the tail of NUM r0 to all zeros Num_Tidy -------- This makes the NUM r0 use as few digits as possible by removing all the leading zeroes Num_UCmp -------- This returns an unsigned comparison between NUM r0 amd NUM r1 in r0 This is the SGN(NUM r0 - NUM r1) Num_Cmp ------- This function returns a signed comparison between NUM r0 and NUM r1 It returns SGN(NUM r0 - NUM r1) Num_ScalarCmp ---------- This function returns a signed comparison between NUM r0 and SCALAR r1 It returns SGN(NUM r0 - NUM r1) Num_ScalarAdd ---------- This adds NUM r0 to (signed int) r1 into NUM r2 Num_ScalarSub ---------- This subtracts NUM r0 - (signed int) r1 into NUM r2 Num_ScalarDiv ---------- This multiplies NUM r0 by (unsigned int) r1 into NUM r2 Num_ScalarMod ---------- This divides NUM r0 by (unsigned int) r1 returning the (unsigned int) remainder in r0 int NUM r2 Num_Mul ------- NUM r0 * NUM r1 -> NUM r2 Num_Div ------- This divides r0 by r1 to give a quotient r2 remainder r3 For "divide and correct algorithm" see Knuth "Seminumerical Algorthms" section 4.3.1 Num_Mod ------- This divides r0 by r1 to remainder r2 Num_MakeSmallPrimes ------------------- This makes all the primes up to r0, and stores them in the RMA pointed to by SmallPrimes. NSmallPrimes is set to the number of primes found. It deallocates any old SmallPrimes array so this may be called more than once. It returns the number of smallprimes found in r0, and a pointer to the array in r1. If there are already more than enough SmallPrimes in the RMA then this will not recalculate. Num_Dump -------- This dumps the number supplied in r0 in Hexadecimal, for debugging Num_ToString ------------ This converts the number pointed to by r0 into a decimal string pointed to by r0. When finished with, this memory should be released with SYS "Num_Heap",3,,r0 It returns the position of the null at the end of the num in r1 Num_Print --------- This prints out the number supplied in r0. It prints no spaces or newlines. Num_Info -------- Expects r0=HeapPointer This provides information on the numbers module Num_FromString -------------- This converts a number in ASCII base 10 pointed to by r1 into the NUM pointed to by r0, until a character which is not 0-9 is reached. If this is a control char then true is returned otherwise false is returned deals with leading "+","-"," " Num_Input --------- Inputs a number in base 10 to NUM r0 Returns r0 flagging ok conversion Num_RndScalar ------------- Expects r0=HeapPointer Produces a random number into r0 from Seed, and updates it, in the range 0-&FFFFFFFF. It works using the algorithm x(n+1)=(1664525 * x(n) + 907633393) MOD 2^32 It is known that the least significant bits are not as random as the most significant bits, so two consecutive random numbers are generated, and combined with EOR after having been rotated by 16 bits with respect to each other. This halves the period. Reference; Knuth, Seminumerical Algorithms 3.3 p102 p84 This is Line 26, with the best spectral co-efficients for a MOD 2^32 generator, listed in Knuth. This form is especially easy to calculate with the ARMs mul instruction. The A term is calculated as the nearest prime to (1/2-1/3*SQR(3))*2^32. Num_SetSeed ----------- Expects r0=HeapPointer This sets the internal 32-bit random number generator to the seed supplied in r1 Num_Rnd ------- This generates a random number 0 <= rnd < r0 to r1 Num_Gcd ------- This finds the gcd(r0,r1) -> r2 using Euclid's algoritm Num_Sqr ------- This takes the square root of r0 to r1 It uses a second order Newton-Raphson convergence. This doubles the number of significant figures on each iteration. Square root of a negative number if returned as 0. It returns the number of iterations in r0 Num_Pow ------- This finds r0 to the power of r1 to r2 Num_PowMod ---------- This finds r0 to the power of r1 MOD r2 to r3 Num_Factorial ------------- This finds NUM r0 factorial to NUM r1 if r0<=1 then r1=0 Num_SmallFactorN ---------------- This sees whether NUM r0 has any small factors. It requires Num_MakeSmallPrimes to have been called. It tests r1 small primes. It returns either the factor found, or 0 to show none found in r0 Num_SmallFactor --------------- This sees whether NUM r0 has any small factors. It requires Num_MakeSmallPrimes to have been called. It tests all the small primes that have been made. It returns either the factor found, or 0 to show none found in r0 Num_Inv ------- This finds r2 such that r0*r2 MOD r1=r3 AND r3=GCD(r0,r1) so if GCD(r0,r1)=1 (for instance if r1 is prime) then this will compute the inverse MOD r1. This is adapted from Knuth; Seminumerical Algorithms 4.5.2 pp325 Num_FermatTest -------------- This finds whether NUM r0 is a prime with respect to SCALAR r1, using the fermat test. That is r0 is not a prime if r1^(r0-1) MOD r0 <> 1 IF r1^(r0-1) MOD r0=1 THEN r0 may be a prime. r1 must be prime for the test to work. IE this test returns false if r0 is not prime, or true if r0 might be prime. This test is less reliable than Num_ProbablyPrime and takes about the same time to execute. Num_ProbablyPrime ----------------- This tests whether r0 is prime or not. The function returns (in r0) false if the number is not prime, and true if the number is probably prime. The algorithm will return true with r0 composite with a probability of less than 0.25. This algorithm is as in Knuth - Seminumerical algorithms 4.5.4P ----------------------------------------------------------------------------- EXAMPLE ------- In the spirit of an illustration is worth 1000 words, here is an example BASIC program using the numbers module. It tries to find any primes of the form 111...111 (these are called rep-units). REM Check to see we have the numbers module *RMEnsure Numbers 0.0 RMLoad Numbers *RMEnsure Numbers 0.0 Error 1 Numbers module not found REM put aside some BASIC memory for the Numbers Heap HeapSize=64*1024 DIM Numbers HeapSize REM Initialise the Heap SYS "Num_HeapInit",Numbers,HeapSize TO hp%,zero%,one%,two% REM Make some small primes for finding small factors SYS "Num_MakeSmallPrimes",5000 TO a% PRINT ;a%;" small primes found" REM This is the number of times we will run Num_ProbablyPrime to be sure REM a repunit is prime timestotest=20 REM Make the NUM repunit% and start it off at 1 SYS "Num_Init",hp% TO repunit% SYS "Num_Set",repunit%,1 FOR n%=2 TO 100000 REM repunit=10*repunit+1, ie add 1 digit to the repunit SYS "Num_ScalarMul",repunit%,10,repunit% SYS "Num_ScalarAdd",repunit%,1,repunit% PRINT "."; REM try to find a small factor of repunit% SYS "Num_SmallFactor",repunit% TO composite IF composite=0 THEN composite=timestotest WHILE composite>0 PRINT "|"; REM test the primality of repunit% timestotest times SYS "Num_ProbablyPrime",repunit% TO pprime IF pprime THEN composite-=1 ELSE REM if the number is composite but passes a test, this is unusual IF composite<>timestotest THEN PRINT "Unusual prime: ";FNprint(repunit%) composite=-1 ENDIF ENDWHILE ENDIF REM print out a prime if we have found one IF composite=0 THEN PRINT "R(";n%;") = ";FNprint(repunit%);" is prime" NEXT n% END REM This prints NUM a% returning a null string DEF FNprint(a%): SYS "Num_Print",a%: ="" ----------------------------------------------------------------------------- ERRORS ------ The following errors are unique to the Numbers module. However the numbers Module will return any OS errors (such as OS_Heap errors) unchanged, except for the addition of a prefix of "Numbers: " Numbers module has become corrupted Returned on initialisation if the Numbers code has been changed (say by a virus or by a disk error) Numbers: Unknown Operation Returned when an unimplemented numbers SWI is called Numbers: Escape Returned when the user pressed Escape in a Looping SWI. Numbers: Invalid Heap or NUM Returned when HeapPointer is invalid, or an Invalid pointer to a NUM is given Numbers: Divide by 0 in Num_ScalarDiv Numbers: Divide by 0 in Num_Div Numbers: Addback Failed in Num_Div This error should never occur. If it does, I would be interested to know! Numbers: N > NSmallPrimes in Num_SmallFactorN Returned when you ask for more SmallPrimes than have been calculated. ----------------------------------------------------------------------------- REFERENCES ========== Books I have found useful whilst writing this module "SemiNumerical Algorithms" (2nd Edition) D.E.Knuth, Addison-Wesley "Elementary Number Theory" D.M.Burton, Allyn and Bacon "Cryptography: an introduction to computer security" J.Seberry, Prentice Hall