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KEYNOTE.PS
资源名称：diehard.zip　[点击查看]
上传用户：wzoulong
上传日期：2021-03-05
资源大小：732k
文件大小：136k
源码类别：

PropertySheet

开发平台：

Others

KEYNOTE.PS：源码内容
  1. %!PS-Adobe-2.0
  2. %%Creator: dvipsk 5.58a Copyright 1986, 1994 Radical Eye Software
  3. %%Title: keynote.dvi
  4. %%Pages: 11
  5. %%PageOrder: Ascend
  6. %%BoundingBox: 0 0 596 842
  7. %%DocumentPaperSizes: a4
  8. %%EndComments
  9. %DVIPSCommandLine: dvips keynote
  10. %DVIPSParameters: dpi=300, compressed, comments removed
  11. %DVIPSSource:  TeX output 1996.01.02:1910
  12. %%BeginProcSet: texc.pro
  13. /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
  14. /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
  15. mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
  16. ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
  17. isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
  18. hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
  19. TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
  20. forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
  21. /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
  22. /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{
  23. /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N
  24. string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
  25. end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
  26. /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]
  27. N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
  28. length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
  29. 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
  30. get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
  31. dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
  32. /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
  33. /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
  34. 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
  35. setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
  36. .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N
  37. /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
  38. gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp
  39. add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add
  40. /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
  41. dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1
  42. adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
  43. idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
  44. putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval
  45. adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg}
  46. {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{
  47. adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2
  48. chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{]
  49. }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
  50. length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{
  51. cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
  52. 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
  53. add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
  54. /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
  55. known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
  56. /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
  57. 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
  58. 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
  59. {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
  60. getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
  61. ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
  62. RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
  63. false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
  64. round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
  65. rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
  66. {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
  67. B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
  68. 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
  69. p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
  70. a}B /bos{/SS save N}B /eos{SS restore}B end
  71. %%EndProcSet
  72. TeXDict begin 39158280 55380996 1000 300 300 (keynote.dvi)
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  462. E end
  463. %%EndProlog
  464. %%BeginSetup
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  466. TeXDict begin
  467. %%PaperSize: a4
  468. %%BeginPaperSize: a4
  469. a4
  470. %%EndPaperSize
  472. %%EndSetup
  473. %%Page: 1 1
  474. 1 0 bop 0 -41 a Fk(Th)o(e)15 b(follo)o(win)o(g)c(i)q(s)j(bas)q(e)q(d)h
  475. (on)f(t)n(h)o(e)h(K)o(eynot)o(e)g(Addre)q(ss,)h(Compu)o(t)o(er)e
  476. (Science)i(an)o(d)e(St)o(a)o(t)o(i)q(st)o(ics:)19 b(16t)n(h)14
  477. b(Symp)q(os)q(iu)o(m)0 9 y(on)k(t)n(h)o(e)h(In)o(t)o(erf)q(ace,)g(A)o
  478. (t)n(lan)o(t)o(a,)f(1984.)29 b(It)19 b(ap)o(p)q(eare)q(d)f(in)g
  479. Fj(The)h(Pr)n(o)n(c)n(e)n(e)n(dings)f Fk(of)f(t)n(h)o(e)i(Conference,)i
  480. (pu)n(bli)q(sh)o(e)q(d)c(b)o(y)0 59 y(Els)q(evier)e(Pre)q(ss.)387
  481. 167 y Fi(A)h(Curren)o(t)g(View)f(of)i(Ran)o(dom)e(Nu)o(m)m(b)q(er)g
  482. (Gen)o(era)o(t)o(ors)741 217 y Fk(George)f(Marsaglia)62
  483. 267 y(Th)o(e)g(a)o(bilit)o(y)d(t)o(o)i(gen)o(era)o(t)o(e)h(sa)o(t)o(i)q
  484. (sf)q(act)o(ory)e(s)q(equence)q(s)k(of)c(ran)o(dom)f(n)n(u)o(m)n(b)q
  485. (ers)i(i)q(s)g(on)o(e)g(of)f(t)n(h)o(e)i(k)o(ey)f(links)f(b)q(et)o(w)o
  486. (een)0 317 y(Compu)o(t)o(er)20 b(Science)i(an)o(d)e(St)o(a)o(t)o(i)q
  487. (st)o(ics.)36 b(St)o(an)o(d)o(ard)21 b(m)o(et)n(h)o(o)q(ds)f(m)o(ay)e
  488. (no)j(lon)o(ger)f(b)q(e)h(suit)o(a)o(ble)e(for)h(increas)q(in)o(gly)0
  489. 367 y(so)o(phi)q(st)o(ica)o(t)o(e)q(d)d(us)q(e)q(s,)j(su)o(c)o(h)e(as)f
  490. (in)h(preci)q(s)q(ion)f(Mon)o(t)o(e)g(Carlo)g(st)o(udie)q(s,)i(t)o(e)q
  491. (st)o(in)o(g)g(for)e(pr)q(im)o(e)q(s,)g(com)n(bin)o(a)o(t)o(or)q(ics)f
  492. (or)0 416 y(pu)n(blic)h(encrypt)o(ion)h(sc)o(h)o(em)o(e)q(s.)30
  493. b(Thi)q(s)17 b(art)o(icle)h(d)o(e)q(scr)q(ib)q(e)q(s)h(str)q(in)o(gen)o
  494. (t)g(n)o(ew)f(t)o(e)q(st)o(s)h(for)f(whic)o(h)f(st)o(an)o(d)o(ard)i
  495. (ran)o(dom)0 466 y(n)n(u)o(m)n(b)q(er)10 b(gen)o(era)o(t)o(ors:)18
  496. b(con)o(gruen)o(t)o(ial,)11 b(shift-regi)q(st)o(er)f(an)o(d)g(lagge)q
  497. (d-Fib)q(on)o(acci,)g(giv)o(e)g(p)q(o)q(or)g(re)q(sul)o(t)o(s,)j(an)o
  498. (d)d(d)o(e)q(scr)q(ib)q(e)q(s)0 516 y(n)o(ew)k(m)o(et)n(h)o(o)q(ds)g(t)
  499. n(h)o(a)o(t)g(pass)h(t)n(h)o(e)f(str)q(in)o(gen)o(t)h(t)o(e)q(st)o(s)h
  500. (an)o(d)d(s)q(eem)h(more)f(suit)o(a)o(ble)g(for)g(preci)q(s)q(ion)h
  501. (Mon)o(t)o(e)f(Carlo)h(us)q(e.)0 566 y(1.)32 b(INTR)o(ODUCTION)62
  502. 616 y(Most)17 b(compu)o(t)o(er)f(syst)o(ems)g(h)o(a)o(v)o(e)g(ran)o
  503. (dom)e(n)n(u)o(m)n(b)q(er)i(gen)o(era)o(t)o(ors)i(a)o(v)n(aila)o(bl)o
  504. (e,)c(an)o(d)h(for)h(most)f(purp)q(os)q(e)q(s)j(t)n(h)o(ey)0
  505. 666 y(w)o(or)o(k)h(rem)o(ar)o(ka)o(bly)f(w)o(ell.)33
  506. b(In)o(d)o(ee)q(d,)21 b(a)e(ran)o(dom)e(n)n(u)o(m)n(b)q(er)i(gen)o(era)
  507. o(t)o(or)h(i)q(s)f(m)n(u)o(c)o(h)f(lik)o(e)h(s)q(ex:)29
  508. b(wh)o(en)19 b(it)o(s)h(go)q(o)q(d)e(it)o(s)0 715 y(w)o(on)o(d)o
  509. (erful,)13 b(an)o(d)h(wh)o(en)g(it)o(s)h(bad)e(it)o(s)i(st)o(ill)e
  510. (pret)n(t)o(y)i(go)q(o)q(d.)62 765 y(Bu)o(t)g(m)o(an)o(y)d(of)i(t)n(h)o
  511. (e)g(st)o(an)o(d)o(ard)h(ran)o(dom)d(n)n(u)o(m)n(b)q(er)i(gen)o(era)o
  512. (t)o(ors)h((RNG's))f(are)g(not)g(go)q(o)q(d)f(enough)h(for)g
  513. (increas-)0 815 y(in)o(gly)j(so)o(phi)q(st)o(ica)o(t)o(e)q(d)h(Mon)o(t)
  514. o(e)g(Carlo)g(us)q(e)q(s,)i(su)o(c)o(h)f(as)f(in)g(geom)o(etr)q(ic)e
  515. (proba)o(bilit)o(y)m(,)h(com)n(bin)o(a)o(t)o(or)q(ics,)g(e)q(st)o(im)o
  516. (a)o(t)o(in)o(g)0 865 y(di)q(str)q(ibu)o(t)o(ion)d(fu)o(nct)o(ions,)h
  517. (compar)q(in)o(g)e(st)o(a)o(t)o(i)q(st)o(ical)h(pro)q(ce)q(d)o(ure)q
  518. (s,)j(gen)o(era)o(t)o(in)o(g)e(an)o(d)g(t)o(e)q(st)o(in)o(g)g(for)g
  519. (large)g(pr)q(im)o(e)q(s)e(for)0 915 y(us)q(e)k(in)f(encrypt)o(ion)h
  520. (sc)o(h)o(em)o(e)q(s)g(an)o(d)f(t)n(h)o(e)h(lik)o(e.)25
  521. b(A)o(t)17 b(least,)g(t)n(h)o(a)o(t)g(i)q(s)f(m)o(y)f(curren)o(t)i
  522. (view,)g(an)o(d)f(I)h(will)e(giv)o(e)h(reasons)0 964
  523. y(for)e(it,)f(b)o(y)g(d)o(e)q(scr)q(ibin)o(g)i(n)o(ew,)f(more)f(str)q
  524. (in)o(gen)o(t)h(t)o(e)q(st)o(s)i(t)n(h)o(a)o(t)e(st)o(an)o(d)o(ard)h
  525. (RNG's)e(f)q(ail)f(y)o(et)i(n)o(ew)g(kin)o(ds)g(of)f(gen)o(era)o(t)o
  526. (ors)0 1014 y(pass.)62 1064 y(Th)o(e)j(more)e(str)q(in)o(gen)o(t)i(t)o
  527. (e)q(st)o(s)i(are)d(st)o(ill)g(reason)o(a)o(ble,)g(rela)o(t)o(e)q(d)h
  528. (t)o(o)g(t)n(h)o(e)g(kin)o(ds)f(of)f(ap)o(plica)o(t)o(ions)g(of)g
  529. (RNG's)h(t)n(h)o(a)o(t)0 1114 y(c)o(h)o(eap)10 b(an)o(d)f(f)q(ast)g
  530. (compu)o(t)o(in)o(g)g(p)q(o)o(w)o(er)g(h)o(a)o(v)o(e)g(m)o(ad)o(e)g
  531. (feas)q(ible,)g(oft)o(en)g(callin)o(g)f(for)h(sample)q(s)f(of)h(h)n(u)o
  532. (n)o(dre)q(ds)j(of)c(t)n(h)o(ousan)o(ds)0 1164 y(or)k(millio)o(ns.)j
  533. (Th)o(e)d(kin)o(ds)g(of)f(gen)o(era)o(t)o(ors)j(t)n(h)o(a)o(t)d(pass)i
  534. (t)n(h)o(e)g(more)e(str)q(in)o(gen)o(t)h(t)o(e)q(st)o(s)i(are)e(t)n(h)o
  535. (os)q(e)h(t)n(h)o(a)o(t)f(com)n(bin)o(e)f(s)q(imple,)0
  536. 1213 y(st)o(an)o(d)o(ard)k(gen)o(era)o(t)o(ors)g(in)f(v)n(ar)q(ious)e
  537. (w)o(ays.)62 1263 y(Most)17 b(of)e(t)n(h)o(e)h(st)o(an)o(d)o(ard)g(RNG)
  538. f('s)g(pro)q(d)o(u)o(ce)h(a)g(s)q(equence)h(of)e(elem)o(en)o(t)o(s)h(b)
  539. o(y)f(m)o(eans)g(of)g(a)g(lin)o(ear)h(transform)o(a-)0
  540. 1313 y(t)o(ion)e(on)h(som)o(e)e(alge)o(braic)i(stru)o(ct)o(ure.)22
  541. b(Three)16 b(m)o(et)n(h)o(o)q(ds)f(domin)o(a)o(t)o(e:)j(con)o(gruen)o
  542. (t)o(ial,)c(shift-regi)q(st)o(er)h(an)o(d)g(lagge)q(d-)0
  543. 1363 y(Fib)q(on)o(acci.)j(Th)o(e)q(s)q(e)13 b(will)f(b)q(e)h(calle)q(d)
  544. g Fj(simple)f Fk(RNG's;)g(d)o(et)o(ails)h(an)o(d)f(pro)o(p)q(ert)o(ie)q
  545. (s)i(will)e(b)q(e)h(su)o(mm)o(ar)q(ize)q(d)e(in)h(s)q(ect)o(ions)0
  546. 1413 y(b)q(elo)o(w.)21 b(Most)16 b(example)q(s)e(of)g(s)q(imple)f(gen)o
  547. (era)o(t)o(ors)k(pass)f(st)o(an)o(d)o(ard)f(t)o(e)q(st)o(s)i(for)e(ran)
  548. o(domn)o(e)q(ss,)g(su)o(c)o(h)h(as)f(t)n(h)o(os)q(e)h(en)n(u-)0
  549. 1463 y(m)o(era)o(t)o(e)q(d)g(in)f(Kn)n(u)o(t)n(h's)j(V)m(olu)o(m)o(e)c
  550. (2)i([6],)f(bu)o(t)h(m)o(ay)f(f)q(ail)f(sp)q(ect)o(acularly)i(on)g(on)o
  551. (e)g(or)g(more)f(of)h(t)n(h)o(e)h(n)o(ew,)f(str)q(in)o(gen)o(t)0
  552. 1512 y(t)o(e)q(st)o(s)j(d)o(e)q(scr)q(ib)q(e)q(d)f(b)q(elo)o(w.)26
  553. b(M)o(et)n(h)o(o)q(ds)18 b(t)n(h)o(a)o(t)f(com)n(bin)o(e)e(t)o(w)o(o)i
  554. (of)f(t)n(h)o(e)h(s)q(imple)e(gen)o(era)o(t)o(ors)j(do)f(m)n(u)o(c)o(h)
  555. f(b)q(et)n(t)o(er)i(on)e(t)n(h)o(e)0 1562 y(str)q(in)o(gen)o(t)f(t)o(e)
  556. q(st)o(s.)62 1612 y(Aft)o(er)d(su)o(mm)o(ary)c(d)o(e)q(scr)q(ipt)o
  557. (ions)k(of)e(t)n(h)o(e)h(t)n(hree)h(most)f(common)c(s)q(imple)i(RNG's)g
  558. (:)17 b(con)o(gruen)o(t)o(ial,)10 b(shift-regi)q(st)o(er)0
  559. 1662 y(an)o(d)j(lagge)q(d-Fib)q(on)o(acci)f((wit)n(h)i(more)e(d)o(et)o
  560. (ail)h(on)g(t)n(h)o(e)h(la)o(t)n(t)o(er)g(t)o(w)o(o,)e(as,)h(in)g(spit)
  561. o(e)g(of)g(b)q(e)q(in)o(g)g(cons)q(id)o(ere)q(d)i(b)o(y)e(v)n(ar)q
  562. (ious)0 1712 y(re)q(s)q(earc)o(h)o(ers)18 b(for)d(t)n(h)o(e)g(past)h
  563. (25)e(y)o(ears,)i(t)n(h)o(ey)f(are)g(not)g(as)g(wid)o(ely)g(kno)o(wn)f
  564. (as)h(con)o(gruen)o(t)o(ial)g(RNG's),)e(I)i(will)f(giv)o(e)0
  565. 1761 y(som)o(e)d(t)n(h)o(eoret)o(ical)h(just)o(i14ca)o(t)o(ion)g(for)
  566. f(com)n(binin)o(g)f(s)q(imple)g(gen)o(era)o(t)o(ors,)k(t)n(h)o(en)f(d)o
  567. (e)q(scr)q(ib)q(e)g(n)o(ew,)f(str)q(in)o(gen)o(t)h(t)o(e)q(st)o(s)h(t)n
  568. (h)o(a)o(t)0 1811 y(s)q(imple)d(gen)o(era)o(t)o(ors)16
  569. b(f)q(ail)c(bu)o(t)j(com)n(bin)o(a)o(t)o(ion)c(gen)o(era)o(t)o(ors)k
  570. (pass.)0 1885 y(2.)36 b(SIMPLE)15 b(GENERA)m(TORS:)g(CONGR)o(UENTIAL)g
  571. (Th)o(e)q(s)q(e)h(gen)o(era)o(t)o(ors)h(us)q(e)f(a)f(lin)o(ear)g
  572. (transform)o(a)o(t)o(ion)e(on)0 1934 y(t)n(h)o(e)i(r)q(in)o(g)e(of)g
  573. (re)q(d)o(u)o(ce)q(d)j(re)q(s)q(id)o(ue)q(s)f(of)e(som)o(e)g(mo)q(d)o
  574. (ulus)g(m,)f(t)o(o)i(pro)q(d)o(u)o(ce)h(a)e(s)q(equence)j(of)d(in)o(t)o
  575. (egers:)515 2023 y Fh(x)539 2029 y Fg(1)558 2023 y Fh(;)7
  576. b(x)601 2029 y Fg(2)619 2023 y Fh(;)g(x)662 2029 y Fg(3)679
  577. 2023 y Fh(;)g(:)g(:)g(:)12 b Fk(wit)n(h)i Fh(x)877 2029
  578. y Ff(n)911 2023 y Fk(=)e Fh(ax)1001 2029 y Ff(n)p Fe(00)p
  579. Fg(1)1075 2023 y Fk(+)d Fh(b)i Fk(mo)q(d)g Fh(m:)0 2111
  580. y Fk(Th)o(ey)k(are)h(t)n(h)o(e)f(most)g(wid)o(ely)f(us)q(e)q(d)i
  581. (RNG's,)e(an)o(d)h(t)n(h)o(ey)g(w)o(or)o(k)h(rem)o(ar)o(ka)o(bly)d(w)o
  582. (ell)h(for)h(most)f(purp)q(os)q(e)q(s.)23 b(Bu)o(t)16
  583. b(for)0 2160 y(som)o(e)e(purp)q(os)q(e)q(s)j(t)n(h)o(ey)e(are)g(not)g
  584. (sa)o(t)o(i)q(sf)q(act)o(ory;)f(p)q(oin)o(t)o(s)h(in)g(n-space)g(pro)q
  585. (d)o(u)o(ce)q(d)h(b)o(y)f(con)o(gruen)o(t)o(ial)f(RNG's)g(f)q(all)f(on)
  586. 0 2210 y(a)i(la)o(t)n(t)o(ice)h(wit)n(h)g(a)f(h)n(uge)h(u)o(nit)g(cell)
  587. g(v)o(olu)o(m)o(e,)e Fh(m)747 2195 y Ff(n)p Fe(00)p
  588. Fg(1)812 2210 y Fk(,)i(compare)q(d)f(t)o(o)h(t)n(h)o(e)g(u)o(nit)g
  589. (cell)g(v)o(olu)o(m)o(e)e(of)h(1)g(t)n(h)o(a)o(t)h(w)o(ould)f(b)q(e)0
  590. 2260 y(exp)q(ect)o(e)q(d)k(f)q(rom)c(truly)i(ran)o(dom)e(in)o(t)o
  591. (egers.)29 b(Det)o(ails)16 b(are)i(in)e([9,10].)26 b(Con)o(gruen)o(t)o
  592. (ial)17 b(RNG's)f(p)q(erform)g(w)o(ell)g(on)0 2310 y(m)o(an)o(y)c(of)h
  593. (t)n(h)o(e)i(str)q(in)o(gen)o(t)f(t)o(e)q(st)o(s)i(d)o(e)q(scr)q(ib)q
  594. (e)q(d)g(b)q(elo)o(w,)c(bu)o(t)j(not)e(on)h(all)f(of)g(t)n(h)o(em.)0
  595. 2383 y(3.)32 b(SIMPLE)14 b(GENERA)m(TORS:)f(SHIFT-REGISTER)62
  596. 2433 y(Th)o(e)q(s)q(e)20 b(are)f(bas)q(e)q(d)h(on)f(viewin)o(g)f(t)n(h)
  597. o(e)i(bit)o(s)f(of)f(a)h(compu)o(t)o(er)g(w)o(ord)g(as)f(t)n(h)o(e)i
  598. (elem)o(en)o(t)o(s)f(of)f(a)h(bin)o(ary)f(v)o(ect)o(or,)0
  599. 2483 y(t)n(h)o(en)g(us)q(in)o(g)f(it)o(era)o(t)o(e)q(s)h(of)e(a)h(lin)o
  600. (ear)g(transform)o(a)o(t)o(ion)e(t)o(o)i(gen)o(era)o(t)o(e)h(a)f(s)q
  601. (equence)i(of)d(bin)o(ary)h(v)o(ect)o(ors,)i(an)o(d)e(h)o(ence)0
  602. 2533 y(compu)o(t)o(er)h(w)o(ords,)g(t)n(h)o(a)o(t)f(m)o(ay)f(b)q(e)h
  603. (in)o(t)o(erpret)o(e)q(d)i(as)f(a)f(s)q(equence)i(of)e(u)o(niform)f
  604. (ran)o(dom)g(in)o(t)o(egers.)29 b(In)17 b(t)o(erms)h(of)0
  605. 2583 y(v)o(ect)o(ors)d(an)o(d)d(m)o(a)o(tr)q(ice)q(s,)g(t)n(h)o(e)i(s)q
  606. (equence)h(e)e(i)q(s)f Fh(14)r(;)7 b(14)r(T)s(;)g(14)r(T)877
  607. 2567 y Fg(2)898 2583 y Fh(;)g(14)r(T)972 2567 y Fg(3)990
  608. 2583 y Fh(;)g(:)g(:)g(:)k Fk(wit)n(h)i Fh(14)j Fk(a)c(1)c
  609. Fd(02)f Fh(n)13 b Fk(bin)o(ary)g(v)o(ect)o(or,)g(T)g(an)g
  610. Fh(n)7 b Fd(02)h Fh(n)0 2632 y Fk(bin)o(ary)13 b(m)o(a)o(tr)q(ix)f
  611. (of)h(0's)g(an)o(d)g(1's,)g(all)g(ar)q(it)n(hm)o(et)o(ic)f(mo)q(d)o
  612. (ulo)f(2)i(an)o(d)h(addit)o(ion)e(of)h(bin)o(ary)g(v)o(ect)o(ors)j(t)n
  613. (h)o(e)e(exclus)q(iv)o(e-or)890 2757 y(1)p eop
  614. %%Page: 2 2
  615. 2 1 bop 0 -41 a Fk(o)o(p)q(era)o(t)o(ion)17 b((d)o(e)q(s)q(ign)o(a)o
  616. (t)o(e)q(d)i Fd(10)p Fk())f(of)g(t)n(h)o(e)h(t)o(w)o(o)e(corre)q(sp)
  617. q(on)o(din)o(g)j(compu)o(t)o(er)e(w)o(ords.)31 b(Shift-regi)q(st)o(er)
  618. 18 b(gen)o(era)o(t)o(ors)i(are)0 9 y(som)o(et)o(im)o(e)q(s)11
  619. b(calle)q(d)j(T)m(a)n(usw)o(ort)n(h)o(e)g(gen)o(era)o(t)o(ors.)19
  620. b(Th)o(e)14 b(bin)o(ary)f(m)o(a)o(tr)q(ix)e(T)j(i)q(s)f(usually)f(c)o
  621. (h)o(os)q(en)j(so)f(t)n(h)o(a)o(t)f(t)n(h)o(e)h(pro)q(d)o(u)o(ct)0
  622. 59 y Fh(14)r(T)k Fk(m)o(ay)10 b(b)q(e)i(pro)q(d)o(u)o(ce)q(d)h(wit)n
  623. (h)f(s)q(imple)d(compu)o(t)o(er)j(o)o(p)q(era)o(t)o(ions.)k(A)c(go)q(o)
  624. q(d)g(c)o(h)o(oice)g(i)q(s)f Fh(T)17 b Fk(=)12 b(()p
  625. Fh(I)d Fk(+)c Fh(R)1499 43 y Ff(s)1516 59 y Fk()()p
  626. Fh(I)k Fk(+)c Fh(L)1640 43 y Ff(t)1655 59 y Fk(),)12
  627. b(wh)o(ere)0 108 y Fh(R)i Fk(i)q(s)g(t)n(h)o(e)g(m)o(a)o(tr)q(ix)e(t)n
  628. (h)o(a)o(t)j(transforms)e(ev)o(ery)i(v)o(ect)o(or)h Fh(14)e
  629. Fk(=)f(()p Fh(b)936 114 y Fg(1)954 108 y Fh(;)7 b(b)991
  630. 114 y Fg(2)1009 108 y Fh(;)g(:)g(:)g(:)e(;)i(b)1120 114
  631. y Ff(n)p Fe(00)p Fg(1)1184 108 y Fh(;)g(b)1221 114
  632. y Ff(n)1243 108 y Fk())14 b(in)o(t)o(o)g Fh(14)r Fk(()p
  633. Fh(b)1416 114 y Fg(2)1435 108 y Fh(;)7 b(b)1472 114 y
  634. Fg(3)1490 108 y Fh(;)g(:)g(:)g(:)e(;)i(b)1601 114 y Ff(n)1623
  635. 108 y Fh(;)g Fk(0).)18 b(Th)n(us)0 158 y Fh(R)c Fk(i)q(s)f(all)g(0's)h
  636. (except)h(for)f(1's)f(on)h(t)n(h)o(e)h(pr)q(incipal)e(sup)q(er-diagon)o
  637. (al;)f(L)i(i)q(s)g(all)e(0's)i(except)h(for)f(1's)g(on)f(t)n(h)o(e)i
  638. (pr)q(incipal)0 208 y(su)n(b)q(diagon)o(al.)62 258 y(Th)o(en)g
  639. Fh(14)r(T)20 b Fk(=)13 b Fh(14)r Fk(()p Fh(I)h Fk(+)c
  640. Fh(R)431 243 y Ff(s)448 258 y Fk()()p Fh(I)k Fk(+)c
  641. Fh(L)582 243 y Ff(t)597 258 y Fk())15 b(m)o(ay)e(b)q(e)h(eas)q(ily)g
  642. (pro)q(d)o(u)o(ce)q(d)i(b)o(y)f(formin)o(g)d Fh(14)k
  643. Fd( )c Fh(14)g Fd(10)e Fh(14)r(R)1511 243 y Ff(s)1530
  644. 258 y Fk(,)k(wit)n(h)h(a)f(logical)0 308 y(r)q(igh)o(t-shift-s)19
  645. b(an)o(d)g(a)h Fd(10)p Fk(,)g(follo)o(w)o(e)q(d)e(b)o(y)i
  646. Fh(14)j Fd( )e Fh(14)16 b Fd(10)d Fh(14)r(L)898
  647. 293 y Ff(t)933 308 y Fk(wit)n(h)20 b(a)f(logical)f(left-shift-t)h(an)o
  648. (d)h(anot)n(h)o(er)g Fd(10)p Fk(.)36 b(Th)o(e)0 357
  649. y(m)o(axim)m(u)o(m)10 b(p)q(oss)q(ible)j(p)q(er)q(io)q(d)g(of)f(a)h
  650. (shift-regi)q(st)o(er)g(gen)o(era)o(t)o(or)i(i)q(s)d(2)1040
  651. 342 y Ff(n)1070 357 y Fd(00)c Fk(1,)k(t)n(h)o(e)i(n)n(u)o(m)n(b)q(er)
  652. f(of)f(non-n)n(ull)g(1)c Fd(02)g Fh(n)k Fk(bin)o(ary)0
  653. 407 y(v)o(ect)o(ors.)24 b(F)m(or)15 b(su)o(c)o(h)h(a)f(gen)o(era)o(t)o
  654. (or,)h(an)o(y)f(non-n)n(ull)f(init)o(ial)f((s)q(ee)q(d))k(v)o(ect)o
  655. (or)g(m)o(ay)c(b)q(e)j(us)q(e)q(d,)g(a)f(d)o(e)q(s)q(ira)o(ble)g(pro)o
  656. (p)q(ert)o(y)0 457 y(for)i(gen)o(era)o(t)o(ors)j(t)n(h)o(a)o(t)e(allo)o
  657. (w)e(s)q(et)n(t)o(in)o(g)j(t)n(h)o(e)f(s)q(ee)q(d)h(v)n(alue,)f(whic)o
  658. (h)f(t)o(o)h(t)n(h)o(e)h(us)q(er)g(b)q(ecom)o(e)q(s)e(an)o(y)h
  659. (non-zero)g(in)o(t)o(eger.)0 507 y(Ma)o(tr)q(ix)11 b(t)n(h)o(eory)i
  660. (pro)o(vid)o(e)q(s)g(an)f(easy)h(m)o(eans)f(t)o(o)g(c)o(h)o(aract)o(er)
  661. q(ize)i(shift-regi)q(st)o(er)e(s)q(equence)q(s)j(of)d(m)o(axim)n(al)c
  662. (p)q(er)q(io)q(d;)k(for)0 557 y(an)i(easy)g(pro)q(of,)f(s)q(ee)i([11].)
  663. 0 606 y Fc(THEOREM)d(1.)p Fj(L)n(et)f(T)g(b)n(e)g(a)g(nonsingular)g
  664. Fh(n)q Fd(02)q Fh(n)g Fj(binary)g(matrix.)17 b(In)11
  665. b(or)n(der)f(that)h(the)g(se)n(quenc)n(e)h Fh(14)r(;)7
  666. b(14)r(T)s(;)g(14)r(T)1713 591 y Fg(2)1733 606 y
  667. Fh(;)g(:)g(:)g(:)0 656 y Fj(have)18 b(p)n(erio)n(d)g
  668. Fk(2)246 641 y Ff(n)279 656 y Fd(00)12 b Fk(1)17 b
  669. Fj(for)g(every)h(non-nul)r(l)g(binary)g(se)n(e)n(d)f(ve)n(ctor)h
  670. Fh(14)r Fj(,)g(it)f(is)h(ne)n(c)n(essary)f(and)i(su16cient)e(that)h
  671. (the)0 706 y(matrix)c(T)h(have)g(or)n(der)g Fk(2)404
  672. 691 y Ff(n)435 706 y Fd(00)10 b Fk(1)k Fj(in)h(the)g(gr)n(oup)g(of)g
  673. (nonsingular)g Fh(n)10 b Fd(02)f Fh(n)15 b Fj(binary)g(matric)n(es.)0
  674. 780 y Fk(4.)j(SIMPLE)c(GENERA)m(TORS:)f(LA)o(GGED-FIBONA)o(CCI)62
  675. 829 y(Th)o(e)q(s)q(e)j(gen)o(era)o(t)o(ors)g(us)q(e)f(an)f(init)o(ial)e
  676. (s)q(et)j(of)f(elem)o(en)o(t)o(s)g Fh(x)939 835 y Fg(1)958
  677. 829 y Fh(;)7 b(x)1001 835 y Fg(2)1018 829 y Fh(;)g(:)g(:)g(:)e(;)i(x)
  678. 1135 835 y Ff(r)1167 829 y Fk(an)o(d)14 b(t)o(w)o(o)g(\lags")f
  679. Fh(r)i Fk(an)o(d)f Fh(s)h Fk(wit)n(h)f Fh(r)f(>)g(s)p
  680. Fk(.)0 879 y(Su)o(cce)q(ss)q(iv)o(e)18 b(elem)o(en)o(t)o(s)d(are)h(gen)
  681. o(era)o(t)o(e)q(d)h(b)o(y)e(t)n(h)o(e)i(recurs)q(ion,)f(for)f
  682. Fh(n)f(>)h(r)q Fk(:)21 b Fh(x)1181 885 y Ff(n)1218 879
  683. y Fk(=)15 b Fh(x)1289 885 y Ff(n)p Fe(00)p Ff(r)1363
  684. 879 y Fd(05)c Fh(x)1419 885 y Ff(n)p Fe(00)p Ff(s)1482
  685. 879 y Fk(,)16 b(wh)o(ere)h Fd(05)e Fk(i)q(s)g(som)o(e)0
  686. 929 y(bin)o(ary)d(o)o(p)q(era)o(t)o(ion.)17 b(Th)o(e)c(init)o(ial)e
  687. ((s)q(ee)q(d))j(elem)o(en)o(t)o(s)f(are)g(compu)o(t)o(er)g(w)o(ords)g
  688. (an)o(d)g(t)n(h)o(e)g(bin)o(ary)f(o)o(p)q(era)o(t)o(ion)g(migh)o(t)f(b)
  689. q(e)0 979 y(+,)p Fd(00)p Fk(,)p Fd(03)f Fk(or)h Fd(10)f
  690. Fk((exclus)q(iv)o(e-or).)18 b(F)m(or)10 b(+)h(or)f
  691. Fd(00)p Fk(,)h(t)n(h)o(e)h Fh(x)p Fk('s)e(migh)o(t)e(b)q(e)j(in)o(t)o
  692. (egers)h(mo)q(d)d(2)1256 964 y Ff(k)1286 979 y Fk(or)i(s)q(in)o(gle-)f
  693. (or)h(dou)n(ble-preci)q(s)q(ion)0 1029 y(reals)i(mo)q(d)d(1.)18
  694. b(F)m(or)11 b Fd(03)p Fk(,)h(o)q(dd)g(in)o(t)o(egers)h(mo)q(d)d(2)698
  695. 1014 y Ff(k)718 1029 y Fk(.)18 b(W)m(e)11 b(d)o(e)q(s)q(ign)o(a)o(t)o
  696. (e)i(su)o(c)o(h)g(a)e(gen)o(era)o(t)o(or)j(lo)q(os)q(ely)d(as)h
  697. (F(r,s,)p Fd(05)p Fk(),)g(al)o(t)n(h)o(ough)0 1078
  698. y(eac)o(h)j(lagge)q(d-Fib)q(on)o(acci)e(gen)o(era)o(t)o(or)j(d)o(ep)q
  699. (en)o(ds)f(on)f(d)o(et)o(ails)g(of)f(t)n(h)o(e)i(part)o(icular)f(bin)o
  700. (ary)f(o)o(p)q(era)o(t)o(ion)g(an)o(d)h(t)n(h)o(e)h(14nit)o(e)0
  701. 1128 y(s)q(et)g(of)e(elem)o(en)o(t)o(s)h(it)f(o)o(p)q(era)o(t)o(e)q(s)i
  702. (on.)62 1178 y(Example)q(s)c(of)h(gen)o(era)o(t)o(ors)i(of)e(m)o(axim)n
  703. (al)d(p)q(er)q(io)q(d)j(are)h(F(l7,5,+))e(or)h(F(17,5,)p
  704. Fd(00)p Fk())g(on)g(in)o(t)o(egers)h(mo)q(d)e(2)1640
  705. 1163 y Ff(k)1660 1178 y Fk(,)h(p)q(er)q(io)q(d)0 1228
  706. y((2)37 1213 y Fg(17)73 1228 y Fd(00)q Fk(1)2)164
  707. 1213 y Ff(k)q Fe(00)p Fg(1)227 1228 y Fk(;)e(F(17,5,)p
  708. Fd(00)p Fk())f(on)h(32-bit)e(reals)j(wit)n(h)f(24)f(bit)g(f)q(ract)o
  709. (ions,)h(su)n(btract)o(ion)g(mo)q(d)o(ulo)e(1,)i(p)q(er)q(io)q(d)f((2)
  710. 1638 1213 y Fg(17)1674 1228 y Fd(00)q Fk(1)2)1765
  711. 1213 y Fg(23)0 1278 y Fk(or)15 b(dou)n(ble)g(preci)q(s)q(ion)f(reals)i
  712. (mo)q(d)o(ulo)d(1,)i(p)q(er)q(io)q(d)f((2)823 1263 y
  713. Fg(17)868 1278 y Fd(00)d Fk(1)2)969 1263 y Fg(55)1004
  714. 1278 y Fk(;)k(F(17,5,)p Fd(03)p Fk())e(on)i(o)q(dd)g(in)o(t)o
  715. (egers)h(mo)q(d)e(2)1623 1263 y Fg(32)1657 1278 y Fk(,)h(p)q(er)q(io)q
  716. (d)0 1327 y((2)37 1312 y Fg(17)82 1327 y Fd(00)c Fk(1)2)183
  717. 1312 y Fg(29)218 1327 y Fk(;)16 b(F(17,5,)p Fd(10)p
  718. Fk(),)e(p)q(er)q(io)q(d)h(2)602 1312 y Fg(17)648 1327
  719. y Fd(00)10 b Fk(1)k(=)h(131071.)22 b(Ot)n(h)o(er)17
  720. b(go)q(o)q(d)e(c)o(h)o(oice)q(s)h(for)f((r,s))h(are)g
  721. ((31,13),(55,24),)0 1377 y((68,33),(97,33),(60)o(7,2)o
  722. (73),(12)o(79,)o(418))11 b(plus)i(m)o(an)o(y)f(ot)n(h)o(ers,)j(som)o
  723. (e)e(of)g(whic)o(h)g(are)i(li)q(st)o(e)q(d)e(in)h(Kn)n(u)o(t)n(h)h([6,)
  724. e(p28].)62 1427 y(A)j(lagge)q(d-Fib)q(on)o(acci)e(gen)o(era)o(t)o(or)j
  725. (i)q(s)e(eas)q(ily)g(programm)o(e)q(d,)e(us)q(in)o(g)i(a)g(circular)h
  726. (li)q(st)e(an)o(d)h(t)o(w)o(o)g(p)q(oin)o(t)o(ers.)23
  727. b(F)m(or)0 1477 y(example,)17 b(a)h(pro)q(ce)q(d)o(ure)i(for)e(an)g
  728. (F(17,5,)p Fd(05)p Fk())e(gen)o(era)o(t)o(or)k(us)q(e)q(s)f(17)e(m)
  729. o(emory)f(lo)q(ca)o(t)o(ions)h Fh(L)p Fk((),)j(init)o(ially)15
  730. b(14lle)q(d)j(as)0 1527 y Fh(L)p Fk((1))12 b(=)g
  731. Fh(x)161 1533 y Fg(17)196 1527 y Fh(;)7 b(L)p Fk((2))k(=)h
  732. Fh(x)375 1533 y Fg(16)410 1527 y Fh(;)7 b(:)g(:)g(:)t(;)g(L)p
  733. Fk((17))k(=)h Fh(x)683 1533 y Fg(1)715 1527 y Fk(an)o(d)h(p)q(oin)o
  734. (t)o(ers)h Fh(I)h Fk(=)d(17,)g Fh(J)j Fk(=)d(5.)18 b(Th)o(en)c(eac)o(h)
  735. g(call)e(t)o(o)i(t)n(h)o(e)g(pro)q(ce)q(d)o(ure)0 1577
  736. y(execu)o(t)o(e)q(s)i(t)n(h)o(e)q(s)q(e)g(instru)o(ct)o(ions:)104
  737. 1654 y Fh(L)p Fk(()p Fh(I)s Fk())c Fd( )f Fh(L)p Fk(()p
  738. Fh(I)s Fk())g Fd(05)e Fh(L)p Fk(()p Fh(J)t Fk())104
  739. 1704 y(ou)o(t)o(pu)o(t)15 b Fh(L)p Fk(()p Fh(I)s Fk())104
  740. 1753 y Fh(I)g Fd( )c Fh(I)i Fd(00)c Fk(1;)k(If)h Fh(I)h
  741. Fk(=)d(0)h(,)h Fh(I)h Fd( )c Fk(17)104 1803 y Fh(J)k
  742. Fd( )c Fh(J)j Fd(00)9 b Fk(1;)k(If)h Fh(J)h Fk(=)d(0,)h
  743. Fh(J)j Fd( )11 b Fk(17)0 1880 y((Decrem)o(en)o(t)o(in)o(g)j(t)n(h)o(e)
  744. g(p)q(oin)o(t)o(ers)h(an)o(d)e(t)o(e)q(st)o(in)o(g)i(on)f(0)f(i)q(s)h
  745. (usually)f(f)q(ast)o(er)h(t)n(h)o(an)g(increm)o(en)o(t)o(in)o(g)f(an)o
  746. (d)g(t)o(e)q(st)o(in)o(g)i(on)f(17.))62 1930 y(Ch)o(aract)o(er)q(iza)o
  747. (t)o(ion)19 b(of)g(m)o(axim)o(al)d(p)q(er)q(io)q(d)j(F(r,s,-))g(gen)o
  748. (era)o(t)o(ors)j(m)o(ay)17 b(b)q(e)j(bas)q(e)q(d)g(on)g(t)n(h)o(e)g(t)n
  749. (h)o(eory)g(of)f(lin)o(ear)0 1980 y(recurs)q(iv)o(e)c(s)q(equence)q(s)h
  750. (of)e(in)o(t)o(egers:)478 2057 y Fh(x)502 2063 y Ff(n)536
  751. 2057 y Fk(=)d Fh(a)601 2063 y Fg(1)620 2057 y Fh(x)644
  752. 2063 y Ff(n)p Fe(00)p Fg(1)718 2057 y Fk(+)e Fh(a)781
  753. 2063 y Fg(2)800 2057 y Fh(x)824 2063 y Ff(n)p Fe(00)p
  754. Fg(2)898 2057 y Fk(+)h Fd(01)d(01)g(01)g Fk(+)j
  755. Fh(a)1061 2063 y Ff(k)1081 2057 y Fh(x)1105 2063 y Ff(n)p
  756. Fe(00)p Ff(k)1183 2057 y Fk(mo)q(d)g Fh(m:)0 2134 y
  757. Fk(Th)o(ere)k(i)q(s)f(ext)o(ens)q(iv)o(e)h(lit)o(era)o(t)o(ure)f(on)g
  758. (t)n(hi)q(s)h(problem,)d(m)n(u)o(c)o(h)h(more)g(ext)o(ens)q(iv)o(e)i(t)
  759. n(h)o(an)g(n)o(ece)q(ssary)h(for)e(our)g(purp)q(os)q(e)q(s,)0
  760. 2184 y(for)20 b(it)h(i)q(s)f(p)q(oss)q(ible)h(t)o(o)g(d)o(ev)o(elo)o(p)
  761. g(a)g(br)q(ief)f(s)q(elf-con)o(t)o(ain)o(e)q(d)g(trea)o(t)o(m)o(en)o(t)
  762. h(for)f(c)o(h)o(aract)o(er)q(izin)o(g)i(lagge)q(d-Fib)q(on)o(acci)0
  763. 2234 y(gen)o(era)o(t)o(ors)d(of)f(m)o(axim)n(al)c(p)q(er)q(io)q(d)j
  764. (for)h(t)n(h)o(e)g(most)f(imp)q(ort)o(an)o(t)f(mo)q(d)o(ulus,)h(2)1207
  765. 2219 y Ff(n)1229 2234 y Fk(,)h(us)q(in)o(g)g(only)e(elem)o(en)o(t)o
  766. (ary)h(m)o(a)o(tr)q(ix)0 2284 y(t)n(h)o(eory)m(.)62 2333
  767. y(F)m(or)h(lagge)q(d-Fib)q(on)o(acci)g(gen)o(era)o(t)o(ors,)i(t)n(h)o
  768. (e)g(in)o(t)o(eger)e(recurs)q(ion)h(re)q(d)o(u)o(ce)q(s)i(t)o(o)d
  769. Fh(x)1308 2339 y Ff(n)1350 2333 y Fk(=)h Fh(x)1425 2339
  770. y Ff(n)p Fe(00)p Ff(r)1502 2333 y Fd(06)12 b Fh(x)1570
  771. 2339 y Ff(n)p Fe(00)p Ff(s)1634 2333 y Fk(,)19 b(bu)o(t)g(t)n(h)o(e)0
  772. 2383 y(t)n(h)o(eory)13 b(for)g(t)n(h)o(e)g(gen)o(eral)g(in)o(t)o(eger)g
  773. (recurs)q(ion)g(mo)q(d)f(2)823 2368 y Ff(n)858 2383 y
  774. Fk(i)q(s)g(just)h(as)g(eas)q(ily)f(e)q(st)o(a)o(bli)q(sh)o(e)q(d,)g(b)o
  775. (y)h(cons)q(id)o(er)q(in)o(g)g(an)f(init)o(ial)0 2433
  776. y(v)o(ect)o(or)i(of)d(in)o(t)o(egers,)i Fh(13)f Fk(=)g(()p
  777. Fh(x)454 2439 y Fg(1)472 2433 y Fh(;)7 b(x)515 2439 y
  778. Fg(2)533 2433 y Fh(;)g(:::;)g(x)631 2439 y Ff(r)646 2433
  779. y Fk())13 b(an)o(d)f(su)o(cce)q(ss)q(iv)o(e)j(v)o(ect)o(ors)f
  780. Fh(13T)s(;)7 b(13T)1214 2418 y Fg(2)1232 2433 y Fh(;)g(13T)1308
  781. 2418 y Fg(3)1326 2433 y Fh(;)g(:)g(:)g(:)j Fk(gen)o(era)o(t)o(e)q(d)k
  782. (b)o(y)e(a)g(m)o(a)o(tr)q(ix)0 2483 y Fh(T)20 b Fk(of)13
  783. b(in)o(t)o(egers.)19 b(Here)c(i)q(s)f(t)n(h)o(e)g(re)q(sul)o(t;)h(for)e
  784. (a)h(pro)q(of)g(s)q(ee)g([11].)0 2533 y Fc(THEOREM)22
  785. b(2.)32 b Fj(L)n(et)19 b(r)f(b)n(e)h(an)g Fh(r)13 b Fd(02)g
  786. Fh(r)20 b Fj(matrix)e(of)h(inte)n(gers,)g(with)f(o)n(dd)i(determinant.)
  787. 30 b(In)20 b(or)n(der)e(that)h(the)0 2583 y(se)n(quenc)n(e)d(of)e(ve)n
  788. (ctors)635 2632 y Fh(13;)7 b(13T)s(;)g(13T)811
  789. 2615 y Fg(2)829 2632 y Fh(;)g(13T)905 2615 y Fg(3)922
  790. 2632 y Fh(;)g(:)g(:)g(:)39 b Fk(mo)q(d)11 b(2)1143 2615
  791. y Ff(n)890 2757 y Fk(2)p eop
  792. %%Page: 3 3
  793. 3 2 bop 0 -41 a Fj(have)17 b(p)n(erio)n(d)g Fk((2)260
  794. -56 y Ff(r)289 -41 y Fd(00)11 b Fk(1)2)390 -56 y Ff(n)p
  795. Fe(00)p Fg(1)471 -41 y Fj(for)17 b(every)g Fh(n)e Fd(25)g
  796. Fk(1)i Fj(and)g(every)g(initial)f(ve)n(ctor)g(of)h(inte)n(gers)g
  797. Fh(13)d Fk(=)i(()p Fh(x)1556 -35 y Fg(1)1574 -41 y
  798. Fh(;)7 b(:)g(:)g(:)e(;)i(x)1691 -35 y Ff(r)1709 -41 y
  799. Fk())17 b Fj(not)0 9 y(al)r(l)g(even,)i(it)e(is)g(ne)n(c)n(essary)h
  800. (and)h(su16cient)e(that)h(T)f(have)h(or)n(der)f Fh(j)i
  801. Fk(=)e(2)1168 -6 y Ff(r)1198 9 y Fd(00)11 b Fk(1)18
  802. b Fj(in)f(the)h(gr)n(oup)g(of)g(non-singular)0 59 y(matric)n(es)c(for)g
  803. (mo)n(d)h(2,)g(and)h(or)n(der)e(2j)h(for)f(mo)n(d)h(4)g(and)h(or)n(der)
  804. e(4j)h(for)f(mo)n(d)i(8.)62 108 y Fk(T)m(o)c(v)o(er)q(ify)g(t)n(h)o(a)o
  805. (t)g(a)h(part)o(icular)e(F(r,s,+))i(or)f(F(r,s,-))h(gen)o(era)o(t)o
  806. (or)g(h)o(as)g(m)o(axim)n(al)c(p)q(er)q(io)q(d)j Fh(j)i
  807. Fk(=)e((2)1552 93 y Ff(r)1577 108 y Fd(00)6 b Fk(1)2)1673
  808. 93 y Ff(n)p Fe(00)p Fg(1)1750 108 y Fk(for)0 158 y(in)o(t)o(egers)16
  809. b(mo)q(d)e(2)270 143 y Ff(n)292 158 y Fk(,)h(on)o(e)g(n)o(ee)q(d)h
  810. (only)e(call)h(a)g(m)o(a)o(tr)q(ix-squar)q(in)o(g)d(rou)o(t)o(in)o(e)k
  811. ((mo)q(d)d(8))i(a)g(few)g(more)g(t)n(h)o(an)g Fh(r)h
  812. Fk(t)o(im)o(e)q(s)e(t)o(o)0 208 y(v)o(er)q(ify)f(t)n(h)o(a)o(t)g
  813. Fh(T)19 b Fk(h)o(as)13 b(ord)o(er)i(j)e(,2j)f(,4j)g(for)h(mo)q(d)e
  814. (2,4,8.)16 b Fj(This)e(wil)r(l)f(b)n(e)h(true)g(only)h(if)e
  815. Fh(H)i Fk(=)d Fh(T)1399 193 y Ff(j)r Fg(+1)1472 208 y
  816. Fj(is)i Fh(T)21 b Fj(mo)n(d)14 b(2,)g(not)h Fh(T)0 258
  817. y Fj(mo)n(d)g(4)g(and)h Fh(H)244 243 y Fg(2)277 258 y
  818. Fj(not)f Fh(T)j Fk(mo)q(d)10 b(8)p Fj(.)62 308 y Fk(Th)o(e)17
  819. b(m)o(a)o(tr)q(ix)d Fh(T)23 b Fk(i)q(s)16 b(a)g(companion)f(m)o(a)o(tr)
  820. q(ix,)f(wit)n(h)i(0's)h(ev)o(erywh)o(ere)h(except)g(for)e(1's)g(on)h(t)
  821. n(h)o(e)g(pr)q(incipal)e(su)n(b-)0 357 y(diagon)o(al)h(an)o(d)i(t)o(w)o
  822. (o)g(1's)g(in)g(t)n(h)o(e)h(ap)o(pro)o(pr)q(ia)o(t)o(e)e(p)q(os)q(it)o
  823. (ions)h(of)f(t)n(h)o(e)i(last)f(colu)o(mn.)30 b(F)m(or)18
  824. b(example,)f(t)n(h)o(e)i(F(3,1,+))0 425 y(gen)o(era)o(t)o(or)c(on)f
  825. (in)o(t)o(egers)g(mo)q(d)f(2)507 410 y Ff(n)543 425 y
  826. Fk(h)o(as)28 b Fh(T)17 b Fk(=)715 392 y Fb(00)759 396
  827. y Fa(0)c(0)g(1)759 425 y(1)g(0)g(0)759 453 y(0)g(1)g(1)867
  828. 392 y Fb(01)886 425 y Fk(.)62 496 y(A)g(few)h(calls)e(t)o(o)i(a)e(m)o
  829. (a)o(tr)q(ix-squar)q(in)o(g)f(rou)o(t)o(in)o(e)i(v)o(er)q(i14e)q(s)h
  830. (t)n(h)o(a)o(t)g Fh(T)1051 480 y Fg(8)1082 496 y Fk(i)q(s)f
  831. Fh(T)k Fk(mo)q(d)11 b(2,)h(not)h Fh(T)18 b Fk(mo)q(d)10
  832. b(4)i(an)o(d)h Fh(T)1650 480 y Fg(16)1699 496 y Fk(i)q(s)f(not)0
  833. 545 y Fh(T)18 b Fk(mo)q(d)10 b(8.)17 b(Since)d(T)f(i)q(s)f(nons)q(in)o
  834. (gular)h(an)o(d)g Fh(j)h Fk(=)d(2)773 530 y Fg(3)800
  835. 545 y Fd(00)d Fk(1)13 b(i)q(s)f(pr)q(im)o(e,)f(t)n(h)o(e)j(ord)o(er)h
  836. (of)d Fh(T)19 b Fk(for)13 b(mo)q(d)o(ulus)f(2)h(i)q(s)f
  837. Fh(j)r Fk(.)18 b(Wh)o(en)c Fh(j)0 595 y Fk(i)q(s)e(comp)q(os)q(it)o(e,)
  838. f(say)h Fh(j)i Fk(=)e(2)413 580 y Fg(55)454 595 y Fd(00)6
  839. b Fk(1,)12 b(it)g(t)o(ak)o(e)q(s)h(a)f(lit)n(t)n(le)h(more)e(w)o(or)o
  840. (k)h(t)o(o)h(v)o(er)q(ify)f(t)n(h)o(e)h(ord)o(er)g(of)f(T.)g(Su)o(cce)q
  841. (ss)q(iv)o(e)i(squar)q(in)o(g)0 645 y(will)i(v)o(er)q(ify)g(t)n(h)o(a)o
  842. (t)h Fh(T)320 630 y Ff(j)r Fg(+1)397 645 y Fk(i)q(s)g
  843. Fh(T)23 b Fk(for)16 b(mo)q(d)g(2)h(an)o(d)g(not)g(T)g(for)f(mo)q(d)g(4)
  844. h(et)o(c.,)h(bu)o(t)g(on)o(e)f(m)n(ust)f(also)h(v)o(er)q(ify)f(t)n(h)o
  845. (a)o(t)i Fh(T)1780 630 y Ff(k)0 695 y Fk(i)q(s)d(not)g
  846. Fh(I)k Fk(mo)q(d)14 b(2)h(for)g(eac)o(h)i Fh(k)e Fk(=)f
  847. Fh(j)r(=p)p Fk(,)i(wit)n(h)f Fh(p)g Fk(ran)o(gin)o(g)h(o)o(v)o(er)f(t)n
  848. (h)o(e)h(pr)q(im)o(e)e(divi)q(sors)h(of)g Fh(j)r Fk(.)23
  849. b(Ev)o(en)16 b(t)n(h)o(en,)g(a)f(s)q(imple)0 745 y(compu)o(t)o(er)f
  850. (program)f(will)f(s)q(erv)o(e)j(t)o(o)g(e)q(st)o(a)o(bli)q(sh|or)e
  851. (refu)o(t)o(e|t)n(h)o(a)o(t)i(a)e(pro)o(p)q(os)q(e)q(d)i(F(r,s,+))g
  852. (or)f(F(r,s,-))g(gen)o(era)o(t)o(or)0 794 y(h)o(as)g(m)o(axim)n(al)c
  853. (p)q(er)q(io)q(d.)62 844 y(T)m(o)17 b(14n)o(d)g(t)n(h)o(e)g(p)q(er)q
  854. (io)q(d)g(of)f(an)h(F(r,s,)p Fd(03)p Fk())g(gen)o(era)o(t)o(or)h(u)
  855. o(n)o(d)o(er)h(m)n(ul)o(t)o(iplica)o(t)o(io)o(n)14 b(of)j(re)q(s)q(id)o
  856. (ue)q(s)h(rela)o(t)o(iv)o(ely)e(pr)q(im)o(e)g(t)o(o)0
  857. 894 y(a)j(mo)q(d)o(ulus)e Fh(m)p Fk(,)j(on)o(e)f(n)o(ee)q(d)h(only)e
  858. (expre)q(ss)j(t)n(h)o(e)f(A)n(b)q(elian)d(group)i(of)f(t)n(h)o(os)q(e)j
  859. (re)q(s)q(id)o(ue)q(s)f(as)f(a)f(direct)i(pro)q(d)o(u)o(ct)g(of)0
  860. 944 y(cyclic)c(groups,)g(t)n(h)o(en)g(cons)q(id)o(er)g(t)n(h)o(e)h
  861. (F(r,s,+))e(gen)o(era)o(t)o(or)i(on)e(t)n(h)o(e)i(exp)q(on)o(en)o(t)o
  862. (s)f(of)f(t)n(h)o(e)i(gen)o(era)o(t)o(ors)g(of)e(t)n(h)o(e)h(cyclic)0
  863. 994 y(groups.)25 b(F)m(or)15 b(example,)f(ev)o(ery)j(o)q(dd)f(in)o(t)o
  864. (eger)g(mo)q(d)f(2)860 979 y Ff(n)898 994 y Fk(h)o(as)h(a)f(u)o(nique)i
  865. (repre)q(s)q(en)o(t)o(a)o(t)o(ion)g(as)f(a)f(pro)q(d)o(u)o(ct)i(()p
  866. Fd(00)p Fk(1))1748 979 y Ff(i)1762 994 y Fk(3)1783
  867. 979 y Ff(j)0 1043 y Fk(wit)n(h)e Fh(i)e Fd(2)g(f)p Fk(0)p
  868. Fh(;)7 b Fk(1)p Fd(g)13 b Fk(an)o(d)i Fh(j)g Fd(2)e(f)p
  869. Fk(1)p Fh(;)7 b Fk(2)p Fh(;)g(:)g(:)g(:)s(;)g Fk(2)626
  870. 1028 y Ff(n)p Fe(00)p Fg(2)691 1043 y Fd(g)p Fk(.)20
  871. b(Th)n(us)c(t)n(h)o(e)f(p)q(er)q(io)q(d)g(of)f(F(r,s,)p
  872. Fd(03)p Fk())h(for)f(o)q(dd)h(in)o(t)o(egers)h(mo)q(d)d(2)1665
  873. 1028 y Ff(n)1702 1043 y Fk(i)q(s)h(t)n(h)o(e)0 1093 y(p)q(er)q(io)q(d)g
  874. (of)f(t)n(h)o(e)h(F(r,s,+))h(gen)o(era)o(t)o(or)g(on)e(in)o(t)o
  875. (egers)i(mo)q(d)d(2)913 1078 y Ff(n)p Fe(00)p Fg(1)978
  876. 1093 y Fk(:)0 1143 y Fj(If)k(the)h(F(r,s,)p Fk(+)p Fj())f(gener)n
  877. (ator)h(has)g(maximal)f(p)n(erio)n(d,)h Fk((2)894 1128
  878. y Ff(r)923 1143 y Fd(00)10 b Fk(1)2)1023 1128 y Ff(n)p
  879. Fe(00)p Fg(1)1088 1143 y Fj(,)17 b(for)f(inte)n(gers)g(mo)n(d)h
  880. Fk(2)1454 1128 y Ff(n)1493 1143 y Fj(then)g(the)g(F(r,s,)p
  881. Fd(03)p Fj())0 1193 y(gener)n(ator)e(on)g(o)n(dd)g(inte)n(gers)g(mo)
  882. n(d)g Fk(2)585 1178 y Ff(n)622 1193 y Fj(has)g(p)n(erio)n(d)g
  883. Fk((2)857 1178 y Ff(r)884 1193 y Fd(00)10 b Fk(1)2)984
  884. 1178 y Ff(n)p Fe(00)p Fg(3)1049 1193 y Fj(.)62 1243
  885. y Fk(Th)o(e)18 b(F(r,s,)p Fd(00)p Fk())f(gen)o(era)o(t)o(ors)i(on)e
  886. (reals)h(are)g(part)o(icularly)e(suit)o(a)o(ble)g(for)h(ran)o(dom)e(n)n
  887. (u)o(m)n(b)q(er)i(su)n(brou)o(t)o(in)o(e)q(s)i(t)n(h)o(a)o(t)0
  888. 1293 y(ret)o(ur)q(n)f(UNI)g(or)g(VNI,)g(con)o(t)o(in)n(uous)g(ran)o
  889. (dom)e(v)n(ar)q(ia)o(ble)q(s)g(on)i([0,1))e(or)i((-1,1).)29
  890. b(Ordin)o(ar)q(ily)m(,)17 b(su)o(c)o(h)h(su)n(brou)o(t)o(in)o(e)q(s)0
  891. 1342 y(gen)o(era)o(t)o(e)h(a)e(ran)o(dom)f(in)o(t)o(eger)i(f)q(rom)d
  892. (som)o(e)i(s)q(et,)h(t)n(h)o(en)h(divid)o(e)e(b)o(y)h(t)n(h)o(e)g
  893. (large)q(st)g(in)o(t)o(eger)g(in)f(t)n(h)o(e)h(s)q(et)h(t)o(o)e(get)h
  894. (t)n(h)o(e)0 1392 y(require)q(d)13 b(UNI)g(or)f(VNI.)g(If)f(t)n(h)o(e)i
  895. (init)o(ial)d(reals)j Fh(x)745 1398 y Fg(1)764 1392 y
  896. Fh(;)7 b(x)807 1398 y Fg(2)824 1392 y Fh(;)g(:)g(:)g(:)e(;)i(x)941
  897. 1398 y Ff(r)971 1392 y Fk(are)12 b(all)f(bin)o(ary)h(f)q(ract)o(ions)g
  898. (of)f(t)n(h)o(e)i(form)e Fh(k)q(=)p Fk(2)1664 1377 y
  899. Fg(24)1698 1392 y Fk(,)h(t)n(h)o(en)0 1442 y(t)n(h)o(e)h(bin)o(ary)f(o)
  900. o(p)q(era)o(t)o(ion)f Fh(x)6 b Fd(05)g Fh(y)13 b Fk(=)f
  901. Fd(f)p Fk(if)g Fh(x)f Fd(25)h Fh(y)i Fk(t)n(h)o(en)f
  902. Fh(x)6 b Fd(00)g Fh(y)14 b Fk(els)q(e)f Fh(x)6 b Fd(00)g
  903. Fh(y)i Fk(+)e(1)p Fd(g)13 b Fk(in)e(t)n(h)o(e)j(F(r,s,)p
  904. Fd(05)p Fk())e(gen)o(era)o(t)o(or)h(will)e(pro)q(d)o(u)o(ce)0
  905. 1492 y(a)j(s)q(equence)j(of)d(reals)i(on)e([0,1),)f(eac)o(h)j(a)e(bin)
  906. o(ary)h(f)q(ract)o(ion)f Fh(k)q(=)p Fk(2)1001 1477 y
  907. Fg(24)1050 1492 y Fk(wit)n(h)h(n)n(u)o(m)o(era)o(t)o(or)g(t)n(h)o(e)g
  908. (in)o(t)o(eger)g(t)n(h)o(a)o(t)g(w)o(ould)f(b)q(e)0 1542
  909. y(pro)q(d)o(u)o(ce)q(d)h(b)o(y)f(t)n(h)o(e)g(corre)q(sp)q(on)o(din)o(g)
  910. i(F(r,s,)p Fd(00)p Fk())e(gen)o(era)o(t)o(or)h(on)e(in)o(t)o(egers)
  911. i(mo)q(d)e(2)1241 1527 y Fg(24)1275 1542 y Fk(.)62 1591
  912. y(Th)n(us)i(t)n(h)o(e)f(F(r,s,)p Fd(05)p Fk())g(gen)o(era)o(t)o(or)
  913. h(us)q(in)o(g)e(t)n(h)o(a)o(t)h(a)o(b)q(o)o(v)o(e)g Fd(05)f
  914. Fk(on)h(32-bit)f(reals)h((h)o(a)o(vin)o(g)f(24-bit)g(f)q(ract)o
  915. (ions))h(pro)q(d)o(u)o(ce)q(s)0 1641 y(real)19 b(UNI's)f(on)h([0,1))e
  916. (wit)n(h)h(p)q(er)q(io)q(d)g((2)639 1626 y Ff(r)670
  917. 1641 y Fd(00)12 b Fk(1)2)772 1626 y Fg(23)825 1641
  918. y Fk(direct)n(ly)m(,)20 b(a)o(v)o(oidin)o(g)c(t)n(h)o(e)k(divi)q(s)q
  919. (ion)c(o)o(p)q(era)o(t)o(ion)h(n)o(ece)q(ssary)k(in)0
  920. 1691 y(con)o(v)o(en)o(t)o(ion)o(al)15 b(gen)o(era)o(t)o(ors.)25
  921. b(Th)o(e)16 b(sam)o(e)f(m)o(et)n(h)o(o)q(d)g(will)g(direct)n(ly)h(pro)q
  922. (d)o(u)o(ce)h(64-bit)d(dou)n(ble)i(preci)q(s)q(ion)f(reals)i(wit)n(h)0
  923. 1741 y(p)q(er)q(io)q(d)d((2)167 1726 y Ff(r)194 1741
  924. y Fd(00)c Fk(1)2)294 1726 y Fg(55)328 1741 y Fk(.)0
  925. 1814 y(Lagge)q(d-Fib)q(on)o(acci)j(Gen)o(era)o(t)o(ors)j(wit)n(h)e
  926. Fd(10)62 1864 y Fk(Th)o(e)g(F(r,s,)p Fd(10)p Fk())h(s)q(equence)g
  927. (st)o(art)o(s)h(wit)n(h)e(an)f(init)o(ial)f(s)q(et)j(of)e(compu)o(t)o
  928. (er)h(w)o(ords)g Fh(x)1303 1870 y Fg(1)1321 1864 y Fh(;)7
  929. b(x)1364 1870 y Fg(2)1382 1864 y Fh(;)g(:)g(:)g(:)e(;)i(x)1499
  930. 1870 y Ff(r)1516 1864 y Fk(,)14 b(t)n(h)o(en)g(gen)o(era)o(t)o(e)q(s)0
  931. 1914 y(su)o(cce)q(ss)q(iv)o(e)19 b(w)o(ords)e(b)o(y)g(t)n(h)o(e)h
  932. (recurs)q(ion)f Fh(x)656 1920 y Ff(n)695 1914 y Fk(=)g
  933. Fh(x)768 1920 y Ff(n)p Fe(00)p Ff(r)844 1914 y Fd(10)12
  934. b Fh(x)912 1920 y Ff(n)p Fe(00)p Ff(s)976 1914 y Fk(.)27
  935. b(Th)o(e)17 b(t)o(e)q(st)o(s)i(d)o(e)q(scr)q(ib)q(e)q(d)f(b)q(elo)o(w)e
  936. (sh)o(o)o(w)h(gen)o(era)o(t)o(ors)0 1964 y(bas)q(e)q(d)c(on)f(t)n(hi)q
  937. (s)g(s)q(equence)j(t)o(o)d(b)q(e)h(amon)o(g)e(t)n(h)o(e)i(w)o(orst)g
  938. (of)e(all)g(gen)o(era)o(t)o(ors,)j(u)o(nle)q(ss)h(t)n(h)o(e)e(lag)e
  939. Fh(r)i Fk(i)q(s)f(som)o(e)f(600)h(or)g(more.)0 2014 y(In)f(addit)o
  940. (ion,)f(t)n(h)o(e)j(m)o(axim)m(u)o(m)8 b(p)q(oss)q(ible)j(p)q(er)q(io)q
  941. (d)g(i)q(s)g(2)819 1998 y Ff(r)841 2014 y Fd(00)t Fk(1,)h(wh)o(a)o(t)
  942. o(ev)o(er)g(t)n(h)o(e)g(w)o(ord)f(s)q(ize,)h(f)q(ar)e(sh)o(ort)i(of)f
  943. (t)n(h)o(e)h(a)o(t)n(t)o(ain)o(a)o(ble)0 2063 y((2)37
  944. 2048 y Ff(r)61 2063 y Fd(00)6 b Fk(1)2)157 2048 y
  945. Ff(n)p Fe(00)p Fg(1)234 2063 y Fk(for)12 b(F(r,s,+),)g(F(r,s,)p
  946. Fd(00)p Fk())h(or)f((2)713 2048 y Ff(r)737 2063 y
  947. Fd(00)6 b Fk(1)2)833 2048 y Ff(n)p Fe(00)p Fg(3)910
  948. 2063 y Fk(for)12 b(F(r,s,)p Fd(03)p Fk())g(wit)n(h)g(w)o(ords)h(of)
  949. e(n)i(bit)o(s.)18 b(Th)o(e)12 b(exclus)q(iv)o(e-or)0
  950. 2113 y(o)o(p)q(era)o(t)o(ion,)18 b Fd(10)p Fk(,)h(i)q(s)f(no)g(f)q
  951. (ast)o(er)h(t)n(h)o(an)g(+)f(or)h Fd(00)f Fk(in)g(most)g(compu)o(t)o
  952. (ers,)i(an)o(d)e(not)g(a)o(v)n(aila)o(ble)e(in)i(m)o(an)o(y)e(high-lev)
  953. o(el)0 2163 y(lan)o(guage)q(s.)i(So,)12 b(t)o(ak)o(en)h(wit)n(h)f(t)n
  954. (h)o(e)i(p)q(o)q(or)e(st)o(a)o(t)o(i)q(st)o(ical)g(p)q(erform)o(ance)g
  955. (an)o(d)g(f)q(ar)g(sh)o(ort)o(er)i(p)q(er)q(io)q(ds,)f(on)o(e)f(w)o(on)
  956. o(d)o(ers)i(wh)o(y)0 2213 y(F(r,s,)p Fd(10)p Fk())h(gen)o(era)o(t)o
  957. (ors)h(h)o(a)o(v)o(e)e(ev)o(er)i(b)q(een)f(giv)o(en)f(s)q(er)q(ious)h
  958. (cons)q(id)o(era)o(t)o(ion.)20 b(Bu)o(t)c(t)n(h)o(ey)f(h)o(a)o(v)o(e.)k
  959. (Calle)q(d)14 b(\gen)o(eralize)q(d)0 2263 y(fee)q(dbac)o(k)h(shift)e
  960. (regi)q(st)o(er)i(gen)o(era)o(t)o(ors")g((GFSR's),)e(t)n(h)o(ey)i(h)o
  961. (a)o(v)o(e)e(b)q(een)i(t)n(h)o(e)f(su)n(b)r(ject)i(of)d(s)q(ev)o(eral)i
  962. (pap)q(ers)f([1,2,4,7].)0 2336 y(5.)50 b(COMBINA)m(TION)15
  963. b(GENERA)m(TORS)62 2386 y(Empir)q(ical)8 b(st)o(udie)q(s)j(sugge)q(st)g
  964. (t)n(h)o(a)o(t)f(com)n(binin)o(g)e(t)o(w)o(o)i(or)g(more)f(s)q(imple)f
  965. (gen)o(era)o(t)o(ors,)k(b)o(y)e(m)o(eans)f(of)h(a)g(con)o(v)o(enien)o
  966. (t)0 2436 y(compu)o(t)o(er)j(o)o(p)q(era)o(t)o(ion)e(su)o(c)o(h)j(as)e
  967. (+,)p Fd(00)p Fk(,)p Fd(03)g Fk(or)h Fd(10)g Fk((exclus)q(iv)o
  968. (e-or),)f(pro)o(vid)o(e)q(s)h(a)g(comp)q(os)q(it)o(e)e(wit)n(h)i(b)q
  969. (et)n(t)o(er)h(ran)o(domn)o(e)q(ss)0 2485 y(t)n(h)o(an)g(e)q(it)n(h)o
  970. (er)h(of)e(t)n(h)o(e)i(comp)q(on)o(en)o(t)o(s.)j(Th)o(ere)d(i)q(s)e(in)
  971. o(t)o(ere)q(st)o(in)o(g)i(t)n(h)o(eoret)o(ical)f(sup)o(p)q(ort)h(for)e
  972. (su)o(c)o(h)i(an)e(obs)q(erv)n(a)o(t)o(ion.)62 2535 y(Let)k
  973. Fh(x)g Fk(b)q(e)g(a)f(ran)o(dom)f(v)n(ar)q(ia)o(ble)g(t)o(akin)o(g)h(v)
  974. n(alue)q(s)g(in)h(a)f(14nit)o(e)h(s)q(et)g Fh(S)r Fk(.)27
  975. b(T)m(o)16 b(14x)h(id)o(eas,)h(let)e Fh(S)j Fk(=)e
  976. Fd(f)p Fk(1)p Fh(;)7 b Fk(2)p Fh(;)g Fk(3)p Fd(g)p Fk(;)15
  977. b(t)n(h)o(e)0 2585 y(t)n(h)o(eory)g(i)q(s)e(eas)q(ily)g(ext)o(en)o(d)o
  978. (e)q(d)j(t)o(o)e(an)o(y)f(14nit)o(e)h(s)q(et.)19 b(Let)14
  979. b(t)n(h)o(e)h(proba)o(bilit)o(y)d(v)o(ect)o(or)j(for)f
  980. Fh(x)f Fk(b)q(e)h(()p Fh(a;)7 b(b;)g(c)p Fk(),)12 b(t)n(h)o(a)o(t)i
  981. (i)q(s,)890 2757 y(3)p eop
  982. %%Page: 4 4
  983. 4 3 bop 0 -41 a Fk(Pr)q(()p Fh(x)11 b Fk(=)h(1))f(=)h
  984. Fh(a;)20 b Fk(Pr)q(()p Fh(x)11 b Fk(=)h(2))f(=)h Fh(b;)20
  985. b Fk(Pr)q(()p Fh(x)11 b Fk(=)h(3))f(=)h Fh(c)p Fk(.)62
  986. 9 y(W)m(e)f(s)q(eek)g(t)n(h)o(e)h(u)o(niform)d(di)q(str)q(ibu)o(t)o
  987. (ions)h(on)h Fh(S)r Fk(,)h(wit)n(h)e(proba)o(bilit)o(y)g(v)o(ect)o(or)i
  988. (()1220 -8 y Fg(1)p 1220 -1 17 2 v 1220 23 a(3)1241
  989. 9 y Fh(;)1265 -8 y Fg(1)p 1265 -1 V 1265 23 a(3)1286
  990. 9 y Fh(;)1310 -8 y Fg(1)p 1310 -1 V 1310 23 a(3)1331
  991. 9 y Fk().)17 b(Let)11 b Fh(16)r Fk(()p Fh(x)p Fk())g(b)q(e)g(t)n
  992. (h)o(e)h(di)q(st)o(ance)0 59 y(b)q(et)o(w)o(een)20 b(t)n(h)o(e)h(di)q
  993. (str)q(ibu)o(t)o(ion)e(of)g Fh(x)g Fk(an)o(d)g(t)n(h)o(e)i(u)o(niform)d
  994. (di)q(str)q(ibu)o(t)o(ion,)i(d)o(e14n)o(e)q(d)h(as)e(t)n(h)o(e)i(di)q
  995. (st)o(ance)f(b)q(et)o(w)o(een)g(t)n(h)o(e)0 108 y(corre)q(sp)q(on)o
  996. (din)o(g)15 b(proba)o(bilit)o(y)d(v)o(ect)o(ors:)636
  997. 200 y Fh(16)r Fk(()p Fh(x)p Fk())g(=)f Fd(jj)p Fk(()p
  998. Fh(a;)c(b;)g(c)p Fk())h Fd(00)h Fk(()989 172 y(1)p
  999. 989 190 21 2 v 989 229 a(3)1015 200 y Fh(;)1038 172 y
  1000. Fk(1)p 1038 190 V 1038 229 a(3)1064 200 y Fh(;)1088 172
  1001. y Fk(1)p 1088 190 V 1088 229 a(3)1113 200 y())p Fd(jj)p
  1002. Fh(;)0 288 y Fk(wh)o(ere)18 b Fd(jj)d Fk(i)q(s)h(t)n(h)o(e)i(enclos)q
  1003. (er)g(for)e(an)o(y)h(v)o(ect)o(or)g(norm:)22 b Fh(L)871
  1004. 294 y Fg(1)890 288 y Fh(;)7 b(L)937 294 y Fg(2)956 288
  1005. y Fh(;)g(L)1003 294 y Fe(1)1054 288 y Fk(or)17 b(an)o(y)f
  1006. Fh(L)1217 294 y Ff(p)1236 288 y Fk(.)26 b(Let)17 b Fh(y)i
  1007. Fk(b)q(e)e(som)o(e)e(ot)n(h)o(er)j(ran)o(dom)0 338 y(v)n(ar)q(ia)o(ble)
  1008. d(on)h Fh(S)r Fk(,)i(in)o(d)o(ep)q(en)o(d)o(en)o(t)f(of)f
  1009. Fh(x)p Fk(,)h(wit)n(h)f(proba)o(bilit)o(y)f(v)o(ect)o(or)j(()p
  1010. Fh(r)o(;)7 b(s;)g(t)p Fk().)25 b(F)m(urt)n(h)o(ermore,)17
  1011. b(sup)o(p)q(os)q(e)h(w)o(e)f(h)o(a)o(v)o(e)f(a)0 388
  1012. y(bin)o(ary)d(o)o(p)q(era)o(t)o(ion)g Fd(17)h Fk(on)g
  1013. Fh(S)r Fk(,)g(wit)n(h)g(o)o(p)q(era)o(t)o(ion)f(t)o(a)o(ble)g(formin)o
  1014. (g)f(a)h(La)o(t)o(in)g(square,)i(say)796 517 y Fd(17)p
  1015. 837 535 2 60 v 41 w Fk(1)42 b(2)f(3)p 775 536 250 2 v
  1016. 796 578 a(1)p 837 596 2 60 v 41 w(3)h(1)f(2)796 638 y(2)p
  1017. 837 656 V 41 w(1)h(2)f(3)796 698 y(3)p 837 716 V 41 w(2)h(3)f(1)62
  1018. 782 y(Th)o(en)19 b(w)o(e)f(m)o(ay)e(us)q(e)j(t)n(h)o(e)f(bin)o(ary)g(o)
  1019. o(p)q(era)o(t)o(ion)f Fd(17)h Fk(t)o(o)g(form)e(a)i(n)o(ew)g(ran)o
  1020. (dom)e(v)n(ar)q(ia)o(ble,)h Fh(z)j Fk(=)f Fh(x)12 b Fd(17)f
  1021. Fh(y)q Fk(,)20 b(an)o(d)d(t)n(h)o(e)0 832 y(di)q(str)q(ibu)o(t)o(ion)c
  1022. (of)g Fh(z)j Fk(will)d(b)q(e)h(\clos)q(er")g(t)o(o)g(u)o(niform)f(t)n
  1023. (h)o(an)h(t)n(h)o(a)o(t)g(of)f(e)q(it)n(h)o(er)i Fh(x)f
  1024. Fk(or)f Fh(y)c Fk(:)0 917 y Fc(THEOREM)17 b(3.)28 b Fh(16)r
  1025. Fk(()p Fh(x)9 b Fd(17)g Fh(y)q Fk())k Fd(24)e Fh(16)r
  1026. Fk(()p Fh(x)p Fk())j Fj(and)h Fh(16)r Fk(()p Fh(x)9
  1027. b Fd(17)g Fh(y)q Fk())k Fd(24)e Fh(16)r Fk(()p
  1028. Fh(y)q Fk().)62 1028 y(Th)o(e)j(pro)q(of)g(i)q(s)f(s)q(imple)f(for)h
  1029. (t)n(h)o(e)i(a)o(b)q(o)o(v)o(e)e Fh(S)k Fk(an)o(d)d Fd(17)p
  1030. Fk(,)f(an)o(d)g(it)h(gen)o(eralize)q(s)h(in)e(an)h(ob)o(vious)f(w)o(ay)
  1031. m(.)641 1124 y(Pr()p Fh(x)d Fd(17)f Fh(y)k Fk(=)f(1))f(=)h
  1032. Fh(as)e Fk(+)f Fh(br)h Fk(+)g Fh(ct)641 1184 y Fk(Pr()p
  1033. Fh(x)g Fd(17)f Fh(y)k Fk(=)f(2))f(=)h Fh(at)d Fk(+)h
  1034. Fh(bs)f Fk(+)h Fh(cr)641 1244 y Fk(Pr()p Fh(x)g Fd(17)f
  1035. Fh(y)k Fk(=)f(3))f(=)h Fh(ar)e Fk(+)g Fh(bt)f Fk(+)g
  1036. Fh(cs:)0 1319 y Fk(Th)n(us)15 b(t)n(h)o(e)f(proba)o(bilit)o(y)e(v)o
  1037. (ect)o(or)j(for)f Fh(z)g Fk(=)d Fh(x)e Fd(17)g Fh(y)16
  1038. b Fk(i)q(s)563 1473 y(()p Fh(a;)7 b(b;)g(c)p Fk())691
  1039. 1440 y Fb(00)730 1415 y Fh(s)47 b(t)42 b(r)730 1475
  1040. y(r)h(s)k(t)735 1535 y(t)41 b(r)i(s)893 1440 y Fb(01)924
  1041. 1473 y Fk(=)12 b(()p Fh(a;)7 b(b;)g(c)p Fk())p Fh(M)r(;)20
  1042. b Fk(say)m(.)0 1616 y(Th)o(en,)14 b(s)q(ince)g(()242
  1043. 1600 y Fg(1)p 242 1607 17 2 v 242 1631 a(3)264 1616 y
  1044. Fh(;)287 1600 y Fg(1)p 287 1607 V 287 1631 a(3)309 1616
  1045. y Fh(;)332 1600 y Fg(1)p 332 1607 V 332 1631 a(3)354
  1046. 1616 y Fk())p Fh(M)i Fk(=)c(()491 1600 y Fg(1)p 491
  1047. 1607 V 491 1631 a(3)513 1616 y Fh(;)536 1600 y Fg(1)p
  1048. 536 1607 V 536 1631 a(3)558 1616 y Fh(;)581 1600 y Fg(1)p
  1049. 581 1607 V 581 1631 a(3)602 1616 y Fk(),)329 1714 y
  1050. Fh(16)r Fk(()p Fh(x)d Fd(17)g Fh(y)q Fk())k(=)e
  1051. Fd(jj)p Fk(()p Fh(a;)c(b;)g(c)p Fk())p Fh(M)12 b Fd(00)e
  1052. Fk(()788 1686 y(1)p 788 1705 21 2 v 788 1743 a(3)814
  1053. 1714 y Fh(;)837 1686 y Fk(1)p 837 1705 V 837 1743 a(3)863
  1054. 1714 y Fh(;)886 1686 y Fk(1)p 886 1705 V 886 1743 a(3)912
  1055. 1714 y())p Fd(jj)h Fk(=)h Fd(jj)p Fk([()p Fh(a;)7 b(b;)g(c)p