Nov 94 Challenge
| Volume Number: | | 10
|
| Issue Number: | | 11
|
| Column Tag: | | Programmer’s Challenge
|
Programmer’s Challenge 
By Mike Scanlin, Mountain View, CA
Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.
The Rules
Here’s how it works: Each month we present a different programming challenge here. First, you write some code that solves the challenge. Second, optimize your code (a lot). Then, submit your solution to MacTech Magazine. We choose a winner based on code correctness, speed, size and elegance (in that order of importance) as well as the postmark of the answer. In the event of multiple equally-desirable solutions, we’ll choose one winner at random (with honorable mention, but no prize, given to the runners up). The prize for each month’s best solution is $50 and a limited-edition “The Winner! MacTech Magazine Programming Challenge” T-shirt (not available in stores).
To help us make fair comparisons, all solutions must be in ANSI compatible C (e.g. don’t use Think’s Object extensions). Use only pure C code. We disqualify any entries with any assembly in them (except for challenges specifically stated to be in assembly). You may call any routine in the Macintosh toolbox (e.g., it doesn’t matter if you use NewPtr instead of malloc). We test entries with the FPU and 68020 flags turned off in THINK C. We time routines with the latest THINK C (with “ANSI Settings”, “Honor ‘register’ first”, and “Use Global Optimizer” turned on), so beware if you optimize for a different C compiler. Limit your code to 60 characters wide. This helps with e-mail gateways and page layout.
We publish the solution and winners for this month’s Programmers’ Challenge two months later. All submissions must be received by the 10th day of the month printed on the front of this issue.
Mark solutions “Attn: Programmers’ Challenge Solution” and send them via e-mail - Internet progchallenge@xplain.com, AppleLink MT.PROGCHAL, CompuServe 71552,174 and America Online MT PRGCHAL. Include the solution, all related files, and your contact info. If you send via snail mail, send a disk with those items on it; see “How to Contact Us” on p. 2.
MacTech Magazine reserves the right to publish any solution entered in the Programming Challenge of the Month. Authors grant MacTech Magazine the non-exclusive right to publish entries without limitation upon submission of each entry. Authors retain copyrights for the code.
Huffman Decoding
Being able to decode a compressed bit stream quickly is important in many applications. This month you’ll get a chance to decode one of the most commonly used compression formats around. Huffman codes are variable length bit strings that represent some other bit string. I’m not going to explain the algorithm here (see any decent book on algorithms for that) but I will explain the format of the symbol table you’ll be given and show you how to decode using it (which is all you need to know to do this Challenge).
The symbol table you are passed consists of an array of elements that look like this:
/* 1 */
typedef struct SymElem {
unsigned short symLength;
unsigned short sym;
unsigned short value;
} SymElem, *SymElemPtr;
where sym is the compressed bit pattern, symLength is the number of bits in sym (from 1 to 16, starting from the least significant bit of sym) and value is the uncompressed output value (16 bits). The symbol table will be sorted smallest to largest first by length and then, within each length, by sym. For example, if you had a table with two SymElems like this:
sym = 3; symLength = 2; value = 0xAAAA
sym = 1; symLength = 3; value = 0xBBBB
and then you were given this compressed bit stream to decode 1100111001001, then the output would be 0xAAAA 0xBBBB 0xAAAA 0xBBBB 0xBBBB because the first two 1 bits are the 2-bit symbol ‘11’ (i.e. 3), and the next 3 bits are the 3-bit symbol ‘001’, and so on.
You will have a chance to create a lookup table in an un-timed init routine but, the amount of memory you can use is variable, from 8K to 256K. Your init routine cannot allocate more than maxMemoryUsage bytes or it will be disqualified (this includes ‘static’ and global data):
void *HuffmanDecodeInit(theSymTable, numSymElems, maxMemoryUsage);
SymElemPtr theSymTable;
unsigned short numSymElems;
unsigned long maxMemoryUsage;
The return value from this init routine will be passed to the actual decode routine (as the privateHuffDataPtr parameter). The decode routine (which is timed) will be called with different sets of compressed data that use the same symbol table:
unsigned long HuffmanDecode(theSymTable, numSymElems, bitsPtr,
numBits, outputPtr, privateHuffDataPtr)
SymElemPtr theSymTable;
unsigned short numSymElems;
char *bitsPtr;
unsigned long numBits;
unsigned short *outputPtr;
void *privateHuffDataPtr;
You can assume that outputPtr points to a buffer large enough to hold all of the uncompressed data. The return value from HuffmanDecode is the actual number of bytes that were stored in that buffer. The input bits are pointed to by bitsPtr and there are numBits of them. The first bit to decode is the most significant bit of the byte pointed to by bitsPtr. TheSymTable and numSyms are the same parameters that were passed to HuffmanDecodeInit.
Two Months Ago Winner
Wow! The competition was really tight for the Erase Scribble Challenge. So tight, in fact, that the 5th place winner was only about 2% slower than the first place winner. But Challenge champion Bob Boonstra (Westford, MA) was able to implement code that was just a tiny bit more efficient than many other highly efficient entries. And, as a bonus, his entry was smaller than all but one of the other entries. Despite his post-publication disqualification from the Factoring Challenge (see below) Bob remains our champion with four 1st place showings (including this one). Congratulations!
Here are the times and code sizes for each entry. Numbers in parens after a person’s name indicate how many times that person has finished in the top 5 places of all previous Programmer Challenges, not including this one:
Name time code+data
Bob Boonstra (11) 1363 598
Ernst Munter (3) 1378 1434
John Schlack 1391 1482
Tom Elwertowski (1) 1394 910
Mark Chavira 1395 1538
Jim Sokoloff 1481 1094
Allen Stenger (7) 1911 988
Marcel Rivard 2233 2048
Joshua Glazer 168100 466
At least one contestant pointed out that this Challenge was not entirely realistic because: (1) in a real eraser situation the hit-test routine would return the point that was hit to the caller and, (2) the caller would be removing points from the scribble as segments were erased, thus removing the ‘completely static scribble’ characteristic of this Challenge. I agree, it would have been more realistic to have a dynamic scribble but I was trying to limit the complexity of the routine (and I was also trying to give clever people a chance to exploit the static nature of the data by using the init routine).
Bob’s routine is well commented so I won’t discuss it here. He chose an almost identical algorithm to everyone else but he implemented it just a touch better than everyone else.
New Factoring Winner
It seems that Bob Boonstra and I both made mistakes during the Factoring Challenge (June 1994 MacTech): Bob made the mistake of incorrectly handling some input values and I made the mistake of not finding them. Many thanks to Jim Lloyd (Mountain View, CA) for finding this bug and narrowing down the set of inputs that make it happen.
The bug only happens if the number to be factored was created from composite primes where one of them has the high bit set and the other one doesn’t. If they’re both set or if they’re both clear then the bug doesn’t happen. In case you’re using Bob’s code, there is a simple fix (thanks, Bob). Change this line:
*prime2Ptr = (x+y)>>1;
to this:
*prime2Ptr = (x>>1) + (y>>1) + 1;
Because of this bug I’m going to have to retro-actively disqualify Bob’s entry from the Challenge and declare a new winner. However, the new winner is not simply the previous 2nd place winner. The new winner is from a guy whose code was originally sent in a day late (I had to disqualify it) but whose performance is so much better than anyone else who entered (including Bob) that I’m going to allow it to win in the interest of having the best possible factoring code published. It’s about two orders of magnitude faster than the other entries.
So, our new winner is Nick Burgoyne (Berkeley, CA). It turns out that this Challenge was right up Nick’s alley, considering that he has taught factoring math classes in the past. His entry was the only one to use the quadratic sieve algorithm. If you’re interested in learning more about this algorithm then Nick recommends a book by David Bressoud, Factorization and Primality Testing, published by Springer-Verlag in 1989. It covers the underlying mathematics and also gives further references to work on the quadratic sieve. It does not assume an advanced background in math. Nick is willing to discuss factoring with anyone who is interested via e-mail. His internet address is: sbrb@cats.ucsc.edu. Congrats, Nick!
Here’s Bob’s winning solution to the Erase Scribble Challenge, followed by Nick’s winning solution to the Factoring Challenge:
scribble.c
/* EraseScribble Copyright (c) 1994 J Robert Boonstra
Problem statement: Determine whether a square eraser of diameter eraserSize centered at the thePoint intersects any of the points in the data structure theScribble. Note that while the problem statement refers to line segments, the definition of a "hit" means that only the endpoints matter.
Solution strategy: The ideal approach would be to create a simple bitMap during Initialization indicating whether an eraser at a given location intersected theScribble. The bitMap would be created by stamping a cursor image at the location of each point in theScribble. However, a bitMap covering the required maximum bounding box of 1024 x 1024 would require 2^17 bytes, or four times as much storage as the 32K we are allowed to use.
Therefore, this solution has three cases:
1) If the actual bounding box for theScribble passed to the init function fits in the 32K available, we create a bitMap as above and use it directly.
2) Otherwise, we attempt to create a half-scale bitMap, where each bit represents 4 pixels in the image, 2 in .h and 2 in .v. In the PtInScribble function, we ca then quickly determine in most cases when the eraser does not intersect theScribble, and we have to walk the points in theScribble if the bitMap indicates a possible hit.
3) In the event there is not enough storage for a half-scale bitmap, we create a quarter-scale bitmap, where each bit represents 16 pixels in the image, 4 in .h and 4 in .v. Then we proceed as in case 2.
To optimize examination of the points in theScribble when the bitmap is not full scale, we sort the points in the initialization function and store them in the privateScribbleDataPtr. Although this reduces the amount of storage available for the bitmap by ~2K, it improves worst case performance significantly.
Although the init function is not timed for score, we have written it in assembler, in the spirit of the September Challenge.
/* 2 */
*/
#pragma options(assign_registers,honor_register,mc68020)
Typedefs, defines, and prototoypes
#define ushort unsigned short
#define ulong unsigned long
#define kTotalStorage 0x8000
/*
* Layout of privateScribbleData storage:
* OFFSET CONTENT
* ------ -------
* 0: bitMap
* kTotalStorage-kGlobals-4*gNumPoints: sorted points
* kTotalStorage-kGlobals: global data
*
* Globals stored in PrivateScribbleDataPtr
* gNumPoints:number of points in the scribble
* gHOrigin: min scribble h - eraserSize/2
* gVOrigin: min scribble v - eraserSize/2
* gBMHeight: max scribble h - min scribble h + eraserSize
* gBMWidth: max scribble v - min scribble v + eraserSize
* gRowBytes: bitMap rowBytes
* gMode: flag indicating bitmap scale
*/
#define gNumPoints kTotalStorage-2
#define gHOrigin kTotalStorage-4
#define gVOrigin kTotalStorage-6
#define gBMHeight kTotalStorage-8
#define gBMWidth kTotalStorage-10
#define gRowBytes kTotalStorage-12
#define gMode kTotalStorage-14
#define kGlobals 14
#define kSortedPoints kTotalStorage-kGlobals
#define kBitMap 0
#define kHalfscaleBitMap 1
#define kQrtrscaleBitMap -1
typedef struct Scribble {
Point startingPoint;
Point deltaPoints[1];
} Scribble, *ScribblePtr, **ScribbleHndl;
void *EraseScribbleInit(ScribbleHndl theScribble,
unsigned short eraserSize);
Boolean PtInScribble(Point thePoint,
ScribbleHndl theScribble,
unsigned short eraserSize,
void *privateScribbleDataPtr);
PtInScribble
#define eraserH D0
#define eraserV D1
#define eraserSz D2
#define pointCt D6
#define scribblePt D7
Boolean PtInScribble(Point thePoint,
ScribbleHndl theScribble,
unsigned short eraserSize,
void *privateScribbleDataPtr)
{
asm 68020 {
MOVEM.L D6-D7,-(A7)
MOVE.L thePoint,D1
MOVEA.L privateScribbleDataPtr,A0
MOVEQ #0,D0 ; clear high bits for BFTST
MOVE.W D1,D0
SWAP D1
; Return noHit if eraser is outside bounding box defined by gVOrigin,
gHOrigin,
; gVOrigin+gBMHeight, gHOrigin+gBMWIdth. Note that the bounding box
has already
; been expanded by eraserSize/2 in each direction.
SUB.W gVOrigin(A0),D1
BLT @noHit ; v < boundingBox.top
CMP.W gBMHeight(A0),D1
BGT @noHit ; v > boundingBox.bottom
SUB.W gHOrigin(A0),D0
BLT @noHit ; h < boundingBox.left
CMP.W gBMWidth(A0),D0
BGT @noHit ; h > boundingBox.right
; Check eraser against bitMap; return noHit if not set
MOVE.W gRowBytes(A0),D2
TST.W gMode(A0)
BNE.S @testQrtrScaleBitMap
; Full-scale bitMap case; can return Hit if bit is set
MULU.W D1,D2 ; multiple row by rowBytes
ADDA D2,A0 ; A0 points to correct bitMap row
BFTST (A0){D0:1} ; D0 contains bit offset
BNE @hit
noHit:
MOVEQ #0,D0
MOVEM.L (A7)+,D6-D7
UNLK A6 ; save one branch by returning directly
RTS
testQrtrScaleBitMap:
BGT @testHalfScaleBitMap
; Qrtr-scale bitMap case; cannot always return Hit if set
LSR.W #2,D1 ; * adjust for qrtr-scale bitmap *
LSR.W #2,D0 ; * adjust for qrtr-scale bitmap *
MULU.W D1,D2 ; multiple row by rowBytes
ADDA D2,A0 ; A0 points to correct bitMap row
BFTST (A0){D0:1}
BEQ @noHit
BRA.S @TestPoints
testHalfScaleBitMap:
; Half-scale bitMap case; cannot always return Hit if set
LSR.W #1,D1 ; * adjust for half-scale bitmap *
LSR.W #1,D0 ; * adjust for half-scale bitmap *
MULU.W D1,D2 ; multiple row by rowBytes
ADDA.L D2,A0 ; A0 points to correct bitMap row
BFTST (A0){D0:1}
BEQ @noHit
TestPoints:
; bitMap indicates there might be a hit, need to check
MOVEA.L privateScribbleDataPtr,A0
MOVE.W eraserSize,eraserSz
MOVE.L thePoint,eraserH
MOVE.L eraserH,eraserV
SWAP eraserV
LEA kSortedPoints(A0),A1
; Scan sets of 64 points to find a close match
MOVE.W gNumPoints(A0),D7
LSR.W #6,D7 ; numPoints/64
MOVE.W D7,pointCt
LSL.W #8,D7 ; times 4 bytes/pt * 64 pts
SUBA.L d7,A1
SUBQ #1,pointCt
eightLoop:
MOVE.L -(A1),scribblePt
CMP.W eraserH,scribblePt
BLT.S @hLoop
ADDI #65*4,A1
DBRA pointCt,@eightLoop;
; Note that the sorted scribblePts have been stored with
; eraserSize/2 already added in h and v
hLoop:
MOVE.L -(A1),scribblePt
CMP.W eraserH,scribblePt
BLT.S @hLoop
; All points from here on have a true .h component >= eraserH-eraserSize/2.
Now
; need to look at those where .h <= eraserH+eraserSize/2. Because we
already added
; eraserSize/2 to the sorted point values, we compare against eraserH+eraserSz
ADD.W eraserSz,eraserH
ADD.W eraserV,eraserSz
vLoop:
CMP.W eraserH,scribblePt
BGT.S @noHit
; scribblePt.h is in range, now check .v
SWAP scribblePt
CMP.W eraserV,scribblePt
BLT.S @vLoopCheck
SUB.W eraserSz,scribblePt
BLE.S @hit
vLoopCheck:
MOVE.L -(A1),scribblePt
BRA @vLoop;
hit:
MOVEQ #1,D0
MOVEM.L (A7)+,D6-D7
}
}
EraseScribbleInit
void *EraseScribbleInit(ScribbleHndl theScribble,
unsigned short eraserSize)
{
ulong bitMapStorageAvail;
asm 68020 {
MOVEM.L D3-D7/A2,-(A7)
; Allocate storage - use full 32K.
; Note that it is the responsibility of the caller to release this storage.
MOVE.L #kTotalStorage,D0
NewPtr CLEAR
MOVE.L A0,A2 ; A0 is privateDataPtr
; Scan thru the scribble to find min and max in h and v
MOVEA.L theScribble,A1
MOVEA.L (A1),A2 ; A2 = *theScribble
; Initialize mins and maxes to be the starting point
MOVE.L (A2)+,D7 ; current scribble point
MOVEQ #0,D5 ; clear high bits for ADDA.L later
MOVE.W D7,D5 ; D5 = max h
MOVE.W D7,D4 ; D4 = min h
MOVE.W D7,D1 ; D1 = current h
SWAP D7 ; D7 = max v
MOVE.W D7,D6 ; D6 = min
MOVE.W D7,D2 ; D2 = current v
LEA kSortedPoints(A0),A1 ; sorted point storage
; Set up eraserSize/2
MOVE.W eraserSize,D3
LSR.W #1,D3 ; D3 = eraserSize/2
minMaxLoop:
; Store point for subsequent sorting
MOVE.W D1,D0
ADD.W D3,D0
MOVE.W D0,-(A1) ; store current h + eraserSize/2
MOVE.W D2,D0
ADD.W D3,D0
MOVE.W D0,-(A1) ; store current v + eraserSize/2
; Fetch next point
MOVE.L (A2)+,D0 ; fetch deltaPoint
BEQ @minMaxDone
ADD.W D0,D1 ; update current h
CMP.W D1,D5
BGE @noNewHMax
MOVE.W D1,D5 ; store new max h
BRA.S @noNewHMin
noNewHMax:
CMP.W D1,D4
BLE @noNewHMin
MOVE.W D1,D4 ; store new min h
noNewHMin:
SWAP D0
ADD.W D0,D2 ; update current v
CMP.W D2,D7
BGE @noNewVMax
MOVE.W D2,D7 ; store new max v
BRA.S @noNewVMin
noNewVMax:
CMP.W D2,D6
BLE @noNewVMin
MOVE.W D2,D6 ; store new min v
noNewVMin:
BRA.S @minMaxLoop
minMaxDone:
; Calculate number of points
LEA kSortedPoints(A0),A2
MOVE.L A2,D0
SUB.L A1,D0
LSR.W #2,D0
MOVE.W D0,gNumPoints(A0)
; Calculate bitmap storage available
SUBQ.L #8,A1 ; Reserve room for sentinals
MOVE.L A1,D0
SUB.L A0,D0
MOVE.L D0,bitMapStorageAvail
; Calculate origin = min h/v minus eraserSize/2
MOVE.W eraserSize,D0
; Calculate number of columns
SUB.W D3,D4 ; adjust h origin for eraserSize/2
MOVE.W D4,gHOrigin(A0) ; D4 = H Origin
ADD.W D0,D5 ; add eraserSize to width
SUB.W D4,D5
MOVE.W D5,gBMWidth(A0)
ADDQ #7,D5 ; round up to byte level
LSR.W #3,D5
MOVE.W D5,gRowBytes(A0) ; D5 = rowBytes
; Calculate number of rows
SUB.W D3,D6 ; adjust v origin for eraserSize/2
MOVE.W D6,gVOrigin(A0) ; D6 = V Origin
ADD.W D0,D7 ; add eraserSize to height
SUB.W D6,D7
MOVE.W D7,gBMHeight(A0) ; D7 = number of rows
ADDQ #1,D7
; Calculate number of bytes needed to store bitmap.
MULU.W D5,D7
CMP.L bitMapStorageAvail,D7
BLE.S @haveEnoughStorage
; Not enough storage, so we try a half-scale bitMap
MOVEQ #kHalfscaleBitMap,D1
MOVE.W D1,gMode(A0)
; Adjust gRowBytes for half-scale bitMap
MOVE.W gBMWidth(A0),D5
ADDI #15,D5 ; round up to byte level
LSR.W #4,D5
MOVE.W D5,gRowBytes(A0) ; D5 = rowBytes
LSR.W #1,D0 ; adjust eraser size for half-scale bitmap
MOVE.W gBMHeight(A0),D7
ADDQ #1,D7
LSR.W #1,D7
ADDQ #1,D7
MULU.W D5,D7
CMP.L bitMapStorageAvail,D7
BLE.S @haveEnoughStorage
; Still not enough storage, so we set up a qrtr-scale bitMap
MOVEQ #kQrtrscaleBitMap,D1
MOVE.W D1,gMode(A0)
; Adjust gRowBytes for qrtr-scale bitMap
MOVE.W gBMWidth(A0),D5
ADDI #31,D5 ; round up to byte level
LSR.W #5,D5
MOVE.W D5,gRowBytes(A0) ; D5=rowBytes
LSR.W #1,D0 ; adjust eraser size for qrtr-scale bitmap
haveEnoughStorage:
; Create bitMap
MOVEA.L theScribble,A1
MOVEA.L (A1),A2
; Initial location to stamp eraser image in bitmap is the
; startingPoint, minus the bitmap origin, minus eraserSize/2;
MOVE.L (A2)+,D2 ; Fetch startingPoint
MOVE.W D2,D1 ; D1 = current scribble point h
SWAP D2 ; D2 = current scribble point v
SUB.W D4,D1 ; D1 = scribble point h - H origin
SUB.W D3,D1 ; offset h by -eraserSize/2
SUB.W D6,D2 ; D2 = scribble point v - V origin
SUB.W D3,D2 ; offset v by -eraserSize/2
MOVEQ #0,D4 ; clear high bits for BFINS later
MOVE.W D0,D7 ; save eraser width
MOVEQ #-1,D3 ; set bits for insertion using BFINS;
; Loop through all points in the scribble
stampPointLoop:
MOVEA.L A0,A1
MOVE D7,D6 ; D6 = row counter for DBRA smear
MOVE.W D2,D0 ; D0 = v - origin
TST.W gMode(A0)
BEQ @L1
BGT @L0
LSR.W #1,D0 ; adjust for qrtr-scale bitmap
L0: LSR.W #1,D0 ; adjust for half-scale bitmap
L1:
MULU.W D5,D0 ; D0 = (v - origin) * rowBytes
ADDA.L D0,A1 ; A1 = privateStorage - rowOffset
MOVE.W D7,D0
ADDQ #1,D0 ; D0=eraserSize/2+1, the number of
; bits to set in bitMap;
; Loop through eraserSize+1 rows in bitMap for this point
stampRowLoop:
MOVE.W D1,D4 ; D4 = scribble h - origin
TST.W gMode(A0)
BEQ @L3
BGT @L2
LSR.W #1,D4 ; adjust for qrtr-scale bitmap
L2: LSR.W #1,D4 ; adjust for half-scale bitmap
L3:
BFINS D3,(A1){D4:D0};insert mask D3 of width D0
; into bitmap A1 at offset D4
ADDA.L D5,A1 ; increment by rowBytes
DBRA D6,@stampRowLoop
; Fetch next deltaPoint, quit if done
MOVE.L (A2)+,D0
BEQ @bitMapDone
ADD.W D0,D1 ; update current scribble h
SWAP D0
ADD.W D0,D2 ; update current scribble v
BRA.S @stampPointLoop
bitMapDone:
; Sort scribble points for fast lookup. Simple exchange sort will do.
outerSortLoop:
MOVEQ #0,D6 ; exchange flag = no exchanges
LEA kSortedPoints(A0),A2 ; sorted pt storage
MOVE.W gNumPoints(A0),D7 ;outer loop counter
SUBQ #2,D7 ; DBRA adjustment
MOVE.L -(A2),D0 ; D0 = compare value - h
MOVE.L D0,D1
SWAP D1 ; D1 = compare value - v
innerSortLoop:
MOVE.L -(A2),D2
MOVE.L D2,D3
SWAP D3
CMP.W D0,D2 ; primary sort by h
BLT.S @doSwap
BGT.S @noSwap
CMP.W D1,D3 ; secondary sort by v
BGE.S @noSwap
doSwap:
MOVE.L D2,4(A2)
MOVE.L D0,(A2)
MOVEQ #1,D6
BRA.S @testInnerLoop
noSwap:
MOVE.L D2,D0
MOVE.W D3,D1
testInnerLoop:
DBRA D7,@innerSortLoop
TST.W D6
BNE.S @outerSortLoop;
MOVE.L #0x80008000,-(a2) ; Sentinal
; to end point loop
MOVE.L #0x7FFF7FFF,-(a2) ; Sentinal
; to end point loop
MOVE.L A0,D0 ; return ptr to private data
MOVEM.L (A7)+,D3-D7/A2
}
}
Nick’s Factoring Solution
Factor64.c
/* 3 */
// Factor the product N of two primes each ¾ 32 bits
#include "Factor64.h"
// Factor64() is a simplified version of the quadratic
// sieve using multiple polynomials
Factor64
void Factor64(ulong nL,ulong nH,
ulong *p1,ulong *p2) {
register ushort ls,k;
register ulong i,j;
ushort p,s,qi,hi,r,c,sP,sN,ts
ushort b,m,bm,br,m2,S[5];
ulong d,e,sX,sQ,sq;
Int B,C,Q,R,T,X,Y;
ushort *Ptr,*pf,*sf,*lg,*r1,*r2;
ushort *lv,**hm,*hp,**gm,*gp,*gi
ushort *pv,**Xv,*Yv;
// Check whether N is square
N[1] = nL & 0xffff; N[2] = nL >> 16;
N[3] = nH & 0xffff; N[4] = nH >> 16;
k = 4; while (N[k] == 0) k--; N[0] = k;
sq = floor(sqrt(nL + 4294967296.0*nH));
Set(S,sq); Mul(S,S,S);
if (Comp(S,N) == 0) {
*p1 = sq;
*p2 = sq;
return;
}
// Check N for small factors (up to 0x800)
for (k = 0; k < 616; k += 2) {
p = Prm[k];
if (Modq(N,p) == 0) {
Divq(N,p);
*p1 = p;
*p2 = Unset(N);
return;
}
}
// Allocate memory
Ptr = malloc(0x20000);
if (Ptr == 0) {
printf(" malloc failed \n");
exit(0);
}
X = (ushort *) Ptr; Ptr += 20;
Y = (ushort *) Ptr; Ptr += 20;
B = (ushort *) Ptr; Ptr += 20;
C = (ushort *) Ptr; Ptr += 20;
Q = (ushort *) Ptr; Ptr += 20;
R = (ushort *) Ptr; Ptr += 20;
T = (ushort *) Ptr; Ptr += 20;
pf = (ushort *) Ptr; Ptr += 120;
sf = (ushort *) Ptr; Ptr += 120;
lg = (ushort *) Ptr; Ptr += 120;
r1 = (ushort *) Ptr; Ptr += 120;
r2 = (ushort *) Ptr; Ptr += 120;
lv = (ushort *) Ptr; Ptr += 12000;
hm = (ushort **) Ptr; Ptr += 240;
hm[0] = (ushort *) Ptr; Ptr += 14400;
gm = (ushort **) Ptr; Ptr += 240;
gm[0] = (ushort *) Ptr; Ptr += 24000;
pv = (ushort *) Ptr; Ptr += 120;
Yv = (ushort *) Ptr; Ptr += 120;
Xv = (ushort **) Ptr; Ptr += 240;
Xv[0] = (ushort *) Ptr;
for (k = 1; k < 120; k++) {
hm[k] = hm[k-1] + 120;
gm[k] = gm[k-1] + 120 + k;
Xv[k] = Xv[k-1] + 6;
}
/* sieve parameters: ts and qs are cutoffs, b is size of prime base and
m2 is size of sieve*/
k = Bitsize(N);
b = 2 + (5*k)/4;
bm = b + 3;
if (k > 40) m = 50*k + 400;
else m = 100*k - 1600;
m2 = m + m;
if (k > 40) ts = 196 + k/2;
else ts = 11*k - 24 - (k*k)/8;
sq = ceil(sqrt(sq/m));
// Construct the prime base
L0:k = 2;
i = 0;
while (k < b) {
p = Prm[i];
i++;
s = Modq(N,p);
if (qrs(s,p) == 1) {
pf[k] = p;
sf[k] = mrt(s,p);
lg[k] = Prm[i];
k++;
}
i++;
}
pf[1] = 2;
// Construct quadratic polynomials as needed
while (Prm[i] < sq) i += 2;
hi = i;
qi = 0;
L1: do {if (hi < 616) sP = Prm[hi];
else sP = npr(sP);
while ((sP&3) == 1) {
hi += 2;
if (hi < 616) sP = Prm[hi];
else sP = npr(sP);
}
sN = Modq(N,sP);
hi += 2;
} while (qrs(sN,sP) == -1);
Set(T,sN);
Dif(T,T,N);
Divq(T,sP);
d = Modq(T,sP); d *= sP; d += sN;
sX = hrt(d,sP);
Set(X,sX);
e = (ulong) sP*sP;
Set(B,e);
Mulq(B,m); Dif(B,B,X);
Mul(C,B,B); Dif(C,C,N);
Divq(C,sP); Divq(C,sP);
// Do the sieve for current polynoial
if ((B[1] & 1) == 0) {i = 1; j = 0;}
else {i = 0; j = 1;}
if ((N[1] & 7) == 0) ls = 21;
else if ((N[1] & 3) == 0) ls = 14;
else ls = 7;
while (i < m2) {
lv[i] = ls; i += 2;
lv[j] = 0; j += 2;
}
for (k = 2; k < b; k++) {
p = pf[k];
s = sf[k];
e = sX % p;
d = sP % p; d *= d; d %= p; d = inv(d,p);
if (e < s) e += p;
i = e-s; i *= d; i %= p; i = m-i; i %= p;
r1[k] = i;
j = e+s; j *= d; j %= p; j = m-j; j %= p;
r2[k] = j;
if (i > j) {
e = i;
i = j;
j = e;
}
ls = lg[k];
while (j < m2) {
lv[i] += ls; i += p;
lv[j] += ls; j += p;
}
if (i < m2) lv[i] += ls;
}
// Factor polynomial to find rows of matrix hm[qi,k]
for (i = 0; i < m2; i++)
if (lv[i] > ts) {
hp = hm[qi];
d = (ulong) i*sP;
Set(T,d); Mul(Q,T,T); Add(Q,Q,C);
R[0] = B[0]; R[1] = B[1];
R[2] = B[2]; R[3] = B[3];
s = i+i; Mulq(R,s);
if (Comp(Q,R) == 1) hp[0] = 0;
else hp[0] = 1;
Dif(Q,Q,R);
hp[1] = 0;
while ((Q[1] & 1) == 0) {
hp[1] += 1;
Shiftr(Q);
}
k = b;
while (Q[0] > 2) {
k--;
if (k == 1) goto L2;
p = pf[k];
j = i % p;
if (j == r1[k] || j == r2[k]) {
hp[k] = 1;
Divq(Q,p);
while (Modq(Q,p) == 0) {
hp[k] += 1;
Divq(Q,p);
}
}
else hp[k] = 0;
}
sQ = Unset(Q);
while (sQ > 1) {
k--;
if (k == 1) goto L2;
p = pf[k];
j = i % p;
if (j == r1[k] || j == r2[k]) {
hp[k] = 1;
sQ /= p;
while (sQ%p == 0) {
hp[k] += 1;
sQ /= p;
}
}
else hp[k] = 0;
}
while (k > 2) {
k--;
hp[k] = 0;
}
Mulq(T,sP);
Dif(Xv[qi],T,B);
Yv[qi] = sP;
qi += 1;
if (qi == bm) goto L3;
L2:;
}
goto L1;
// Row reduce gm = hm % 2 and find X^2 = Y^2
// (mod N)
L3:k = N[0];
g = (double) N[k];
if (d > 1) g += N[k-1]/65536.0;
if (d > 2) g += N[k-2]/4294967296.0;
if (d > 3) g += 1/4294967296.0;
for (r = 0; r < bm; r++) {
hp = hm[r]; gp = gm[r];
for (c = 0; c < b; c++)
gp[c] = hp[c] & 1;
br = b + r;
for (c = b; c < br;c++) gp[c] = 0;
gp[br] = 1;
}
for (r = 0; r < bm; r++) {
br = b + r;
gp = gm[r];
c = b - 1;
while (gp[c] == 0 && c>0) c--;
if (c > 0 || gp[0] == 1) {
for (i = r+1; i<bm; i++) {
gi = gm[i];
if (gi[c] == 1) {
gi[c] = 0;
for (k = 0; k<c; k++)
gi[k] ^= gp[k];
for (k=b; k<=br; k++)
gi[k] ^= gp[k];
}
}
}
else {
for (i = 1; i < b; i++)
pv[i] = 0;
Set(X,1); Set(Y,1);
for (k = b; k <= br; k++)
if (gp[k] == 1) {
j = k - b;
Mul(X,X,Xv[j]);
if (X[0]>12) Mod(X);
Mulq(Y,Yv[j]);
if (Y[0]>12) Mod(Y);
for (i=1;i<b; i++)
pv[i] += (long) hm[j][i];
}
Mod(X);
k = pv[1] >> 1;
while (k > 15) {
for (i = Y[0]; i > 0; i--)
Y[i+1] = Y[i];
Y[1] = 0;
Y[0] += 1;
k -= 16;
if (Y[0] > 12) Mod(Y);
}
Mulq(Y,1 << k);
for (i = 2; i < b; i++) {
j = pv[i];
if (j == 0) continue;
if (j == 2) Mulq(Y,pf[i]);
else {
T[1] = pf[i];
T[0] = 1;
while (j > 2) {
j -= 2;
Mulq(T,pf[i]);
}
Mul(Y,Y,T);
}
if (Y[0] > 12) Mod(Y);
}
Mod(Y);
Dif(T,X,Y);
Add(R,X,Y);
if (Comp(N,R)<=0) Dif(R,R,N);
if (T[0] == 0 || R[0] == 0)
continue;
*p1 = Gcd(T);
*p2 = Gcd(R);
return;
}
}
bm += 5;
ts -= 2;
goto L0;
}
End of the function Factor64().
// Inverse of b modulo m (Euclid’s algorithm)
inv
ulong inv(ulong b,ushort m) {
register ulong u,v,t,n;
u = 1;
v = 0;
t = 1;
n = m;
for (;;) {
while ((b & 1) == 0) {
v <<= 1;
if (t & 1) t += m;
t >>= 1;
b >>= 1;
}
if (b == 1) break;
if (b > n) {
b -= n;
u += v;
}
else {
n -= b;
v += u;
}
while ((n & 1) == 0) {
u <<= 1;
if (t & 1) t += m;
t >>= 1;
n >>= 1;
}
}
t *= u; t %= m;
return t;
}
// Determine if n is square modulo p (QR algorithm)
qrs
short qrs(ushort n,ushort p) {
register short j;
register ushort n2,n4;
if (n == 1) return 1;
j = 1;
for (;;) {
n2 = n + n;
n4 = n2 + n2;
p %= n4;
while (p > n) {
if (n & 2) j = -j;
if (p > n2) p -= n2;
else p = n2 - p;
}
if (p == 1) return j;
n %= p;
if (n == 1) return j;
}
}
// Square root of n modulo p
mrt
ushort mrt(ushort n,ushort p) {
register ulong q,s;
register long x,u,v;
ulong m;
long t,r;
if (n == 1) return 1;
q = (p+1) >> 1;
m = n;
if ((q & 1) == 0) {
q >>= 1;
s = 1;
while (q > 0) {
if (q & 1) {
s *= m; s %= p;
}
m *= m; m %= p;
q >>= 1;
}
if (s+s > p) s = p-s;
return s;
}
s = 0;
t = n;
while (qrs(t,p) == 1) {
s += 1;
t += 1-s-s;
if (t%p == 0) {
if (s+s > p) s = p-s;
return s;
}
if (t < p) t += p;
}
q >>= 1;
n = t;
r = s;
u = 1;
v = 1;
while (q > 0) {
x = n*u; x %= p; x *= u; x %= p;
t = r*r; t %= p;
if (t >= x) x = t-x;
else x = t-x+p;
u *= r; u %= p; u += u; u %= p;
r = x;
if (q & 1) {
x = n*u; x %= p; x *= v;
x %= p;
t = s*r; t %= p;
if (t >= x) x = t-x;
else x = t-x+p;
v *= r; v %= p; v += s*u; v %= p;
s = x;
}
q >>= 1;
if (s+s > p) s = p-s;
return s;
}
// Square root of n modulo p^2 for p = 3 (mod 4)
hrt
ulong hrt(ulong n,ushort p) {
register ulong s,t,x,y;
ulong m;
m = n%p;
s = m;
y = 1;
t = p-3; t >>= 2;
while (t > 0) {
if (t&1) {
y *= s; y %= p;
}
s *= s; s %= p;
t >>= 1;
}
x = m*y; x %= p;
s = (ulong) p*p;
t = x*x;
if (n < t) n += s;
n -= t; n /= p; n *= y; n %= p;
t = p; t += 1; t >>= 1;
n *= t; n %= p; n *= p; n += x;
if (n+n > s) n = s-n;
return n;
}
// Get next prime
npr
ushort npr(ushort p) {
register ushort d,s,k;
do {p += 2;
s = floor(sqrt(p));
k = 0;
d = 3;
while (d <= s) {
if (p%d == 0) break;
k += 2;
d = Prm[k];
}
}
while (d <= s);
return p;
}
// Addition S = A + B
Add
void Add(Int S,Int A,Int B) {
register ushort *pH, *pL;
register ulong t;
register ushort c,k;
ushort s,dH,dL;
if (A[0] > B[0] ) {
pH = A; dH = A[0]; pL = B; dL = B[0];
}
else {
pH = B; dH = B[0]; pL = A; dL = A[0];
}
if (dL == 0) {
if (S != pH)
for (k=0;k<=dH; k++) S[k]=pH[k];
return;
}
k = 0;
c = 0;
while (k < dL) {
k++;
t = (ulong) pH[k] + pL[k] + c;
if (t >= 0x10000) {
t -= 0x10000;
c = 1;
}
else c = 0;
S[k] = t;
}
while (c == 1 && k < dH) {
k++;
s = pH[k];
if (s == 0xFFFF) {
S[k] = 0;
c = 1;
}
else {
S[k] = s + 1;
c = 0;
}
}
while (k < dH) {
k++;
S[k] = pH[k];
}
if (c == 1) {
dH += 1;
S[dH] = 1;
}
S[0] = dH;
}
// Difference D = |A - B|
Dif
void Dif(Int D,Int A,Int B) {
register ushort *pH, *pL;
register long t;
register ushort c,k;
ushort s,dH,dL;
short e;
k = A[0];
if (k > B[0]) e = 1;
else {
if (k < B[0]) e = -1;
else {
while (A[k]==B[k] && k > 0) k--;
if (k == 0) {
D[0] = 0;
return;
}
if (A[k] > B[k]) e = 1;
else e = -1;
}
}
if (e == 1) {
pH = A; dH = A[0];
pL = B; dL = B[0];
}
else {
pH = B; dH = B[0];
pL = A; dL = A[0];
}
if (dL == 0) {
if (D != pH)
for (k=0;k<=dH;k++) D[k]=pH[k];
return;
}
c = 0;
k = 0;
while (k < dL) {
k++;
t = (long) pH[k] - pL[k] - c;
if (t < 0) {
t += 0x10000;
c = 1;
}
else c = 0;
D[k] = t;
}
while (c == 1 && k < dH) {
k++;
s = pH[k];
if (s == 0) {
D[k] = 0xFFFF;
c = 1;
}
else {
D[k] = s - 1;
c = 0;
}
}
while (k < dH) {
k++;
D[k] = pH[k];
}
k = dH;
while (D[k] == 0 && k > 0) k--;
D[0] = k;
}
// Multiply P = A * B
Mul
void Mul(Int P,Int A,Int B) {
register ushort *pB;
register ulong t;
register ushort s,n,k;
ushort d,j;
if (A[0] == 0 || B[0] == 0) {
P[0] = 0;
return;
}
d = A[0] + B[0];
for (k = 1; k <= d; k++) bufm[k]= 0;
pB = B;
j = 0;
while (j < A[0]) {
j++;
s = A[j];
k = 0;
n = j;
while (k < pB[0]) {
k++;
t = (ulong) s * pB[k];
bufm[n] += t & 0xFFFF;
n++;
bufm[n] += t >> 16;
}
}
k = 1;
while (k < d) {
t = bufm[k];
bufm[k] = t & 0xFFFF;
k++;
bufm[k] += t >> 16;
}
if (bufm[d] == 0) d -= 1;
for (k = 1; k<=d; k++) P[k]=bufm[k];
P[0] = d;
}
// Quick multiply A = A * b
Mulq
void Mulq(Int A,ushort b) {
register ushort *pA;
register ulong t,w,c;
register ushort d,k;
d = A[0];
if (d == 0) return;
pA = A;
if (b == 0) {
pA[0] = 0;
return;
}
c = 0;
k = 0;
while (k < d) {
k++;
t = (ulong) pA[k] * b;
w = (t & 0xFFFF) + c;
c = t >> 16;
if (w >= 0x10000) {
w -= 0x10000;
c += 1;
}
pA[k] = w;
}
if (c > 0) {
d += 1;
pA[d] = c;
}
A[0] = d;
}
// Modulo R = R % N
Mod
void Mod(Int R) {
register ulong t;
register long w;
register ushort k,j;
ushort d,n,tL,tH;
ulong c;
short e;
double f;
d = N[0];
n = R[0];
if (d > n) return;
for (k = 1; k<=n; k++) buf[k]=R[k];
while (n >= d) {
f = buf[n]*65536.0;
if (n > 1) f += buf[n-1];
t = f/g;
e = n - d;
if (e > 0) {
tL = t & 0xFFFF;
tH = t >> 16;
}
else {
tL = t >> 16;
if (tL == 0) {
if (t < 0xFFFF) break;
else {
k = d;
while (buf[k] == N[k]
&& k > 0) k--;
if (k > 0 && buf[k] < N[k])
break;
tL = 1;
}
}
tH = 0;
e = 1;
}
c = 0;
j = e;
for (k = 1; k <= d; k++) {
t = (ulong) N[k]*tL + c;
w = (ulong) buf[j]-(t & 0xFFFF);
t >>= 16;
if (w < 0) {
buf[j] = 0x10000 + w;
t += 1;
}
else buf[j] = w;
j++;
w = (ulong) buf[j] - t;
if (w < 0) {
buf[j] = 0x10000 + w;
c = 0x10000;
}
else {
buf[j] = w;
c = 0;
}
}
if (tH > 0) {
c = 0;
j = e + 1;
for (k = 1; k <= d; k++) {
t = (ulong) N[k]*tH + c;
w = (ulong) buf[j]-(t&0xFFFF);
t >>= 16;
if (w < 0) {
buf[j] = 0x10000 + w;
t += 1;
}
else buf[j] = w;
if (k == d) break;
j++;
w = (ulong) buf[j] - t;
if (w < 0) {
buf[j] = 0x10000 + w;
c = 0x10000;
}
else {
buf[j] = w;
c = 0;
}
}
}
while (buf[n] == 0 && n > 0) n--;
if (n == d && buf[d] < N[d]) break;
}
for (k = 1; k <= n; k++) R[k] = buf[k];
R[0] = n;
}
// Quick modulo A = A % m
Modq
ushort Modq(Int A,ushort m) {
register ulong z,t;
ushort n,e;
n = A[0];
if (n == 0) return 0;
for (e = 1; e <= n; e++) buf[e] = A[e];
while (n > 1) {
e = n - 1;
t = ((ulong) buf[n] << 16) + buf[e];
z = t/m;
t -= z*m;
buf[e] = t;
if (t == 0) do e--;
while (buf[e] == 0 && e > 0);
n = e;
}
return buf[1] % m;
}
// Quick divide A = A / m
Divq
void Divq(Int A,ushort m) {
register ulong z,t;
ushort n,e;
n = A[0];
if (n == 0) return;
for (e = 1; e <= n; e++) {
buf[e] = A[e];
A[e] = 0;
}
while (n > 1) {
e = n - 1;
t = ((ulong) buf[n] << 16) + buf[e];
z = t/m;
if (z < 0x10000) A[e] = z;
else {
A[e] = z & 0xFFFF;
A[n] = z >> 16;
}
t -= z*m;
buf[e] = t;
if (t == 0) do e--;
while (buf[e] == 0 && e > 0);
n = e;
}
n = buf[1];
if (n >= m) A[1] = n/m;
if (A[A[0]] == 0) A[0] -= 1;
}
// Shift right X = X >> 1
Shiftr
void Shiftr(Int X) {
register ulong t;
register ushort i,j,c,d;
d = X[0];
if (d == 0) return;
i = 0;
j = 1;
c = X[1] >> 1;
while (j < d) {
j++;
t = (ulong) X[j] << 15;
t += c;
i++;
X[i] = t & 0xFFFF;
c = t >> 16;
}
if (c == 0) d -= 1;
else X[d] = c;
X[0] = d;
}
// Compare : {+1 0 -1} as {X > Y X == Y X < Y}
Comp
short Comp(Int X,Int Y) {
register ushort d;
d = X[0];
if (d > Y[0]) return 1;
if (d < Y[0]) return -1;
while (X[d] == Y[d] && d > 0) d--;
if (d == 0) return 0;
if (X[d] > Y[d]) return 1;
return -1;
}
// Convert from unsigned long to Int
Set
void Set(Int X, ulong n) {
if (n == 0) {
X[0] = 0;
return;
}
if (n < 0x10000) {
X[0] = 1;
X[1] = n;
return;
}
X[0] = 2;
X[1] = n & 0xffff;
X[2] = n >> 16;
}
// Convert from Int to unsigned long
Unset
ulong Unset(Int X) {
register ulong n;
register ushort d;
d = X[0];
if (d == 0) return 0;
n = (ulong) X[1];
if (d == 1) return n;
return (n + ((ulong) X[2] << 16));
}
// Number of Bits in X
Bitsize
ulong Bitsize(Int X) {
register ushort d,t;
register ulong n;
d = X[0];
if (d == 0) return 0;
n = (ulong) d << 4;
t = 0x8000;
while ((t & X[d]) == 0) {
n -= 1;
t >>= 1;
}
return n;
}
// The greatest common divisor of A and N
Gcd
ulong Gcd(Int A) {
register long k;
for (k = 0; k < 5; k++) buf[k] = N[k];
while ((A[1]&1) == 0) Shiftr(A);
for (;;)
switch (Comp(A,buf)) {
case 1 : Dif(A,A,buf);
while ((A[1]&1) == 0) Shiftr(A);
break;
case -1 : Dif(buf,buf,A);
while ((buf[1]&1) == 0) Shiftr(buf);
break;
case 0 : return Unset(A);
}
}
End of file Factor64.c
Factor64.h
/* 4 */
// Header file for 64 bit factorization program.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define ulong unsigned long
#define ushort unsigned short
typedef ushort * Int;
ulong inv(ulong,ushort);
short qrs(ushort,ushort);
ushort mrt(ushort,ushort);
ulong hrt(ulong,ushort);
ushort npr(ushort);
void Add(Int,Int,Int);
void Dif(Int,Int,Int);
void Mul(Int,Int,Int);
void Mulq(Int,ushort);
void Mod(Int);
ushort Modq(Int,ushort);
void Divq(Int,ushort);
void Shiftr(Int);
short Comp(Int,Int);
void Set(Int,ulong);
ulong Unset(Int);
ulong Bitsize(Int);
ulong Gcd(Int);
ushort buf[20], N[5];
ulong bufm[20];
double g;
// Odd primes below 0x800 (and their 10*logs)
ushort Prm[] = {
3, 11, 5, 17, 7, 20, 11, 24, 13, 26,
17, 29, 19, 30, 23, 32, 29, 34, 31, 35,
37, 37, 41, 38, 43, 38, 47, 39, 53, 40,
59, 41, 61, 42, 67, 43, 71, 43, 73, 43,
79, 44, 83, 45, 89, 45, 97, 46, 101, 46,
103, 46, 107, 46, 109, 46, 113, 47, 127, 48,
131, 48, 137, 49, 139, 49, 149, 50, 151, 50,
157, 50, 163, 50, 167, 51, 173, 51, 179, 51,
181, 51, 191, 52, 193, 52, 197, 52, 199, 52,
211, 53, 223, 54, 227, 54, 229, 54, 233, 54,
239, 54, 241, 54, 251, 55, 257, 55, 263, 55,
269, 55, 271, 56, 277, 56, 281, 56, 283, 56,
293, 56, 307, 57, 311, 57, 313, 57, 317, 57,
331, 58, 337, 58, 347, 58, 349, 58, 353, 58,
359, 58, 367, 59, 373, 59, 379, 59, 383, 59,
389, 59, 397, 59, 401, 59, 409, 60, 419, 60,
421, 60, 431, 60, 433, 60, 439, 60, 443, 60,
449, 61, 457, 61, 461, 61, 463, 61, 467, 61,
479, 61, 487, 61, 491, 61, 499, 62, 503, 62,
509, 62, 521, 62, 523, 62, 541, 62, 547, 63,
557, 63, 563, 63, 569, 63, 571, 63, 577, 63,
587, 63, 593, 63, 599, 63, 601, 63, 607, 64,
613, 64, 617, 64, 619, 64, 631, 64, 641, 64,
643, 64, 647, 64, 653, 64, 659, 64, 661, 64,
673, 65, 677, 65, 683, 65, 691, 65, 701, 65,
709, 65, 719, 65, 727, 65, 733, 65, 739, 66,
743, 66, 751, 66, 757, 66, 761, 66, 769, 66,
773, 66, 787, 66, 797, 66, 809, 66, 811, 66,
821, 67, 823, 67, 827, 67, 829, 67, 839, 67,
853, 67, 857, 67, 859, 67, 863, 67, 877, 67,
881, 67, 883, 67, 887, 67, 907, 68, 911, 68,
919, 68, 929, 68, 937, 68, 941, 68, 947, 68,
953, 68, 967, 68, 971, 68, 977, 68, 983, 68,
991, 68, 997, 69,1009, 69,1013, 69,1019, 69,
1021, 69,1031, 69,1033, 69,1039, 69,1049, 69,
1051, 69,1061, 69,1063, 69,1069, 69,1087, 69,
1091, 69,1093, 69,1097, 70,1103, 70,1109, 70,
1117, 70,1123, 70,1129, 70,1151, 70,1153, 70,
1163, 70,1171, 70,1181, 70,1187, 70,1193, 70,
1201, 70,1213, 71,1217, 71,1223, 71,1229, 71,
1231, 71,1237, 71,1249, 71,1259, 71,1277, 71,
1279, 71,1283, 71,1289, 71,1291, 71,1297, 71,
1301, 71,1303, 71,1307, 71,1319, 71,1321, 71,
1327, 71,1361, 72,1367, 72,1373, 72,1381, 72,
1399, 72,1409, 72,1423, 72,1427, 72,1429, 72,
1433, 72,1439, 72,1447, 72,1451, 72,1453, 72,
1459, 72,1471, 72,1481, 73,1483, 73,1487, 73,
1489, 73,1493, 73,1499, 73,1511, 73,1523, 73,
1531, 73,1543, 73,1549, 73,1553, 73,1559, 73,
1567, 73,1571, 73,1579, 73,1583, 73,1597, 73,
1601, 73,1607, 73,1609, 73,1613, 73,1619, 73,
1621, 73,1627, 73,1637, 74,1657, 74,1663, 74,
1667, 74,1669, 74,1693, 74,1697, 74,1699, 74,
1709, 74,1721, 74,1723, 74,1733, 74,1741, 74,
1747, 74,1753, 74,1759, 74,1777, 74,1783, 74,
1787, 74,1789, 74,1801, 74,1811, 75,1823, 75,
1831, 75,1847, 75,1861, 75,1867, 75,1871, 75,
1873, 75,1877, 75,1879, 75,1889, 75,1901, 75,
1907, 75,1913, 75,1931, 75,1933, 75,1949, 75,
1951, 75,1973, 75,1979, 75,1987, 75,1993, 75,
1997, 75,1999, 76,2003, 76,2011, 76,2017, 76,
2027, 76,2029, 76,2039, 76, 0, 0 };
End of file Factor64.h
