Ciphers thiab neeg soj xyuas
Hnub no Math Corner, Kuv yuav los saib ib lub ntsiab lus uas kuv tau tham hauv National Children's Foundation's Science Camp rau cov menyuam yaus txhua xyoo. Lub hauv paus tab tom nrhiav rau cov menyuam yaus thiab cov tub ntxhais hluas uas nyiam kev tshawb fawb. Koj tsis tas yuav muaj txiaj ntsig zoo heev, tab sis koj yuav tsum muaj "scientific streak." Tsis tas yuav tau qhab nia tsev kawm ntawv zoo heev. Sim nws, koj yuav nyiam nws. Yog tias koj yog ib lub tsev kawm ntawv qib qis lossis tsev kawm theem siab, thov. Feem ntau cov niam txiv lossis tsev kawm ntawv ua cov ntawv ceeb toom, tab sis qhov no tsis yog ib txwm muaj. Nrhiav lub Foundation lub vev xaib thiab nrhiav kom paub.
Muaj ntau thiab ntau tham nyob rau hauv tsev kawm ntawv txog "coding", xa mus rau cov dej num yav tas los hu ua "programming". Qhov no yog ib qho txheej txheem rau cov kws qhia theoretical. Lawv khawb cov txheej txheem qub, muab lawv lub npe tshiab, thiab "kev vam meej" yog ua los ntawm nws tus kheej. Muaj ntau qhov chaw uas muaj qhov tshwm sim cyclical tshwm sim.
Nws tuaj yeem xaus lus tias kuv ntsuas qhov kev ua txhaum cai. Tsis muaj. Nyob rau hauv txoj kev loj hlob ntawm kev vam meej, qee zaum peb rov qab mus rau qhov uas yog, raug tso tseg thiab tam sim no tau rov qab los. Tab sis peb lub ces kaum yog lej, tsis yog philosophical.
Kev koom nrog ib lub zej zog tshwj xeeb kuj txhais tau tias "cov cim ib txwm", kev nyeem ntawv, cov lus thiab cov lus piv txwv. Tus neeg uas kawm lus Polish zoo kawg nkaus "muaj lub thicket loj nyob rau hauv Szczebrzeszyn, ib tug beetle yog buzzing nyob rau hauv lub reeds" yuav tam sim ntawd raug raws li ib tug neeg soj xyuas ntawm ib tug txawv teb chaws lub xeev yog hais tias nws tsis teb lo lus nug ntawm dab tsi woodpecker ua. Tau kawg nws ua tsis taus pa!
Qhov no tsis yog dag xwb. Thaum lub Kaum Ob Hlis Ntuj xyoo 1944, cov neeg German tau pib lawv qhov kev tawm tsam kawg hauv Ardennes ntawm tus nqi loj. Lawv tau tsa cov tub rog uas hais lus Askiv zoo los cuam tshuam kev txav ntawm cov tub rog sib tw, piv txwv li los ntawm kev coj lawv mus rau qhov tsis ncaj ncees lawm ntawm txoj kev sib tshuam. Tom qab ib pliag ntawm kev xav tsis thoob, cov neeg Asmeskas pib nug cov tub rog cov lus nug uas tsis txaus ntseeg, cov lus teb uas yuav pom tseeb rau ib tus neeg los ntawm Texas, Nebraska lossis Georgia thiab inconceivable rau ib tug neeg uas tsis loj hlob nyob rau ntawd. Ignorance ntawm qhov tseeb coj ncaj qha mus rau kev tua.
Rau qhov taw tes. Kuv pom zoo rau cov neeg nyeem phau ntawv los ntawm Lukasz Badowski thiab Zaslaw Adamashek "Labboratory in a Desk Drawer - Mathematics". Qhov no yog ib phau ntawv zoo heev uas qhia tau hais tias kev ua lej yeej muaj txiaj ntsig zoo rau qee yam thiab "kev sim lej" tsis yog lo lus khoob. Nws suav nrog, ntawm lwm yam, piav qhia kev tsim kho ntawm "cardboard enigma" - ib qho khoom siv uas yuav siv sijhawm peb kaum tsib feeb los tsim thiab ua haujlwm zoo li lub tshuab cipher loj. Lub tswv yim nws tus kheej tau paub zoo, cov kws sau ntawv tau hais ua haujlwm zoo nkauj, thiab kuv yuav hloov nws me ntsis thiab qhwv nws hauv cov khaub ncaws lej ntau dua.
hacksaws
Ntawm ib qho ntawm txoj kev ntawm kuv lub zos dacha nyob rau hauv suburbs ntawm Warsaw, lub pavement twb tsis ntev los no dismantled los ntawm "trlinka" - hexagonal paving slabs. Kev caij tsheb tsis xis nyob, tab sis tus ntsuj plig ntawm tus lej zoo siab. Npog lub dav hlau nrog tsis tu ncua (piv txwv li tsis tu ncua) polygons tsis yooj yim. Nws tsuas tuaj yeem yog daim duab peb sab, squares thiab hexagons tsis tu ncua.
Tej zaum kuv tso dag me ntsis nrog kev xyiv fab ntawm sab ntsuj plig, tab sis lub hexagon yog ib daim duab zoo nkauj. Los ntawm nws koj tuaj yeem ua kom muaj kev vam meej encryption ntaus ntawv. Geometry yuav pab tau. Lub hexagon muaj rotational symmetry - nws overlaps nws tus kheej thaum tig los ntawm ntau yam ntawm 60 degrees. Piv txwv li, daim teb tau cim nrog tsab ntawv A nyob rau sab laug fig. 1 Tom qab tig los ntawm lub kaum sab xis no, nws tseem yuav poob rau hauv lub thawv A - thiab tib yam nrog lwm cov ntawv. Yog li cia peb txiav tawm rau lub squares ntawm daim phiaj, txhua tus nrog ib tsab ntawv sib txawv. Peb muab daim phiaj uas tau txais nyob rau hauv txoj kev no rau ntawm ib daim ntawv. Nyob rau hauv qhov dawb rau lub teb, sau rau tsab ntawv ntawm cov ntawv uas peb xav encrypt. Cia peb tig daim ntawv 60 degrees. Rau qhov chaw tshiab yuav tshwm sim - sau rau tsab ntawv tom ntej ntawm peb cov lus.
Rice. 1. Trlinks ntawm kev xyiv fab ntawm kev ua lej.
Ntawm sab xis fig. 1 peb muaj ib cov ntawv encoded li no: "Muaj ib tug loj hnyav chav locomotive ntawm qhov chaw nres tsheb."
Tam sim no me ntsis kev kawm lej yuav los ua ke. Muaj pes tsawg txoj hauv kev tuaj yeem npaj ob tus lej txheeb ze rau ib leeg?
Cov lus nug ruam twg? Rau ob: ib qho rau pem hauv ntej los yog lwm yam.
Zoo. Thiab peb tus lej?
Nws kuj tsis yooj yim rau sau tag nrho cov chaw:
123, 132, 213, 231, 312, 321, XNUMX.
Zoo, nws yog rau plaub! Nws tseem tuaj yeem sau meej meej. Xav txog txoj cai xaj kuv muab tso rau:
1234, 1243, 1423, 4123, 1324, 1342,
1432, 4132, 2134, 2143, 2413, 4213,
2314, 2341, 2431, 4231, 3124, 3142,
3412, 4312, 3214, 3241, 3421, 4321
Thaum tus lej yog tsib, peb tau txais 120 qhov ua tau. Wb hu lawv kev hloov pauv. Tus naj npawb ntawm permutations ntawm n tus lej yog cov khoom 1 2 3 ... n, hu ua muaj zog thiab cim nrog exclamation point: 3!=6, 4!=24, 5!=120. Rau tus lej tom ntej no peb muaj 6!=6. Peb yuav siv qhov no los ua kom peb cov hexagonal cipher shield ntau dua.
Peb xaiv ib tug permutation ntawm cov zauv los ntawm 0 mus rau 5, piv txwv li 351042. Peb hexagonal scrambling disk muaj ib tug khiav ceev nyob rau hauv nruab nrab teb - yog li ntawd nws yuav muab tso rau "nyob rau hauv lub xoom txoj hauj lwm" - ib tug khiav ceev, raws li nyob rau hauv daim duab. 1. Peb muab lub disk ua li no rau ntawm daim ntawv uas peb yuav tsum sau peb daim ntawv qhia, tab sis peb tsis sau tam sim ntawd, tab sis tig peb zaug ntawm 60 degrees (piv txwv li 180 degrees) thiab sau rau tsab ntawv hauv cov teb khoob. Peb rov qab mus rau qhov chaw pib. Peb tig lub xov tooj tsib zaug los ntawm 60 degrees, uas yog, los ntawm tsib "cov hniav" ntawm peb lub xov tooj. Peb luam tawm. Tom ntej no scale txoj hauj lwm yog txoj hauj lwm tig 60 degrees nyob ib ncig ntawm xoom. Qhov thib plaub txoj hauj lwm yog 0 degrees, qhov no yog txoj hauj lwm pib.
Koj puas nkag siab qhov tshwm sim? Peb muaj lub cib fim ntxiv - los ua kom peb "tshuab" nyuaj los ntawm ntau tshaj xya puas zaus! Yog li, peb muaj ob txoj haujlwm ywj pheej ntawm "automaton" - kev xaiv ntawm kab sib chaws thiab kev xaiv ntawm kev hloov pauv. Daim phiaj tuaj yeem xaiv tau hauv 66 = 46656 txoj kev, permutation 720. Qhov no muab 33592320 possibilities. Ntau tshaj 33 lab tus lej cim! Yuav luag me ntsis, vim qee daim phiaj tsis tuaj yeem txiav tawm ntawm daim ntawv.
Hauv qab fig. 1 peb muaj ib co lus sau li no: "Kuv yuav xa plaub parachute rau koj." Nws yog ib qho yooj yim to taub hais tias tus yeeb ncuab yuav tsum tsis txhob pub kom paub txog qhov no. Tab sis nws puas yuav to taub ib qho ntawm no:
LUB SIJ HAWM NTAWM LUB SIJ HAWM
YYLOAKVMDEYCHESH,
txawm nrog kos npe 351042?
Peb tab tom tsim Enigma, German cipher tshuab
Rice. 2. Ib qho piv txwv ntawm kev teeb tsa thawj zaug ntawm peb lub tshuab encryption.
Permutations (AF) (BJ) (CL) (DW) (EI) (GT) (HO) (KS) (MX) (NU) (PZ) (RY).
Raws li kuv twb tau hais lawm, kuv tshuav lub tswv yim ntawm kev tsim lub tshuab cardboard zoo li no rau phau ntawv "Lab hauv Drawer - lej". Kuv "kev tsim kho" txawv me ntsis ntawm qhov muab los ntawm nws cov kws sau ntawv.
Lub tshuab cipher siv los ntawm cov neeg German thaum tsov rog muaj ib txoj hauv kev yooj yim ingeniously, zoo ib yam li peb pom nrog hex cipher. Txhua lub sijhawm tib yam: rhuav tshem txoj haujlwm nyuaj ntawm tsab ntawv mus rau lwm tsab ntawv. Nws yuav tsum hloov tau. Yuav ua li cas thiaj li tswj tau nws?
Cia peb xaiv tsis muaj kev hloov pauv, tab sis ib qho uas muaj qhov ntev 2. Cias muab, ib yam dab tsi zoo li "Gaderipoluk" tau piav qhia ntawm no ob peb lub hlis dhau los, tab sis npog tag nrho cov tsiaj ntawv ntawm cov tsiaj ntawv. Wb pom zoo 24 tsab ntawv - tsis muaj ą, ę, ć, ó, ń, ś, ó, ż, ź, v, q. Muaj pes tsawg permutations? Qhov no yog ib txoj hauj lwm rau cov kawm tiav high school (lawv yuav tsum daws tau tam sim ntawd). Muaj pes tsawg? Ntau ntau? Ob peb txhiab? Yog:
1912098225024001185793365052108800000000 (tseem tsis txhob sim nyeem tus lej no). Muaj ntau txoj hauv kev los teeb tsa txoj haujlwm "zero". Thiab nws tuaj yeem nyuaj.
Peb lub tshuab muaj ob lub voj voog. Ntawm ib ntawm lawv, uas tseem sawv, cov ntawv sau. Nws yog me ntsis zoo li lub xov tooj ntawm lub xov tooj qub, qhov twg koj hu xov tooj los ntawm tig lub xov tooj mus rau txhua txoj kev. Rotary yog qhov thib ob nrog cov qauv xim. Qhov yooj yim tshaj plaws yog muab lawv tso rau ntawm ib lub cork tsis tu ncua siv tus pin. Es tsis txhob cork, koj tuaj yeem siv cov ntawv nyias nyias los yog cov ntawv tuab tuab. Lukasz Badowski thiab Zasław Adamaszek pom zoo muab ob lub discs rau hauv lub thawv CD.
Xav txog tias peb xav encode lo lus ARMATY (Rice. 2i3 ib). Teem lub cuab yeej rau xoom txoj hauj lwm ( xub nce). Tsab ntawv A sib raug rau F. Tig lub voj voog sab hauv ib tsab ntawv mus rau sab xis. Peb muaj tsab ntawv R rau encode, tam sim no nws sib raug rau A. Tom qab kev sib hloov tom ntej, peb pom tias tsab ntawv M sib raug rau U. Kev sib hloov tom ntej (plaub daim duab) muab cov ntawv A - P. Ntawm lub thib tsib dial peb muaj T. - A. Thaum kawg (lub voj voog thib rau) Y – Y Tus yeeb ncuab tej zaum yuav tsis twv tias peb cov CFCFAs yuav txaus ntshai rau nws. Thiab yuav ua li cas "peb" nyeem qhov xa? Lawv yuav tsum muaj tib lub tshuab, tib yam "programmed", uas yog, nrog tib permutation. Lub cipher pib ntawm txoj hauj lwm xoom. Yog li tus nqi ntawm F yog A. Tig lub dial clockwise. Tsab ntawv A yog tam sim no cuam tshuam nrog R. Nws tig lub xov tooj mus rau sab xis thiab hauv qab tsab ntawv U pom M, thiab lwm yam. Tus neeg ua haujlwm cipher khiav mus rau tus thawj: "General, Kuv tab tom qhia, phom tuaj!"
Rice. 3. Lub hauv paus ntsiab lus ntawm kev ua haujlwm ntawm peb daim ntawv Enigma.
| Rice. 3. Lub hauv paus ntsiab lus ntawm kev ua haujlwm ntawm peb daim ntawv Enigma. | ||
Cov muaj peev xwm ntawm txawm xws li ib tug primitive Enigma yog amazing. Peb tuaj yeem xaiv lwm qhov tso zis permutations. Peb tuaj yeem - thiab muaj ntau lub sijhawm ntawm no - tsis yog los ntawm ib qho "serif" tsis tu ncua, tab sis nyob rau hauv ib qho kev hloov pauv txhua hnub, zoo ib yam li hexagon (piv txwv li, thawj peb tsab ntawv, tom qab xya, tom qab, yim, plaub ... .. lwm..).
Koj twv tau li cas?! Thiab tseem rau Polish lej (Marian Reevski, Henry Zigalski, Jerzy Ruzicki: koj puas xav tau ntau tus thwjtim?) tau tshwm sim. Cov ntaub ntawv tau li no yog invaluable. Yav dhau los, lawv muaj qhov tseem ceeb sib npaug rau keeb kwm ntawm peb kev tiv thaiv. Vaclav Serpinski i Stanislav Mazurkevichuas ua txhaum txoj cai ntawm cov tub rog Lavxias teb sab xyoo 1920. Lub intercepted cable muab Piłsudski lub sijhawm los ua tus nto moo maneuver los ntawm tus dej Vepsz.
Kuv nco qab Vaslav Sierpinski (1882-1969). Nws zoo li ib tug lej uas lub ntiaj teb sab nraud tsis muaj nyob. Nws tsis tuaj yeem tham txog nws txoj kev koom tes hauv kev yeej xyoo 1920 ob qho tib si rau kev ua tub rog thiab ... rau kev nom kev tswv (cov tub ceev xwm ntawm Polish People's Republic tsis nyiam cov neeg tiv thaiv peb ntawm Soviet Union).
Fig. 4. Permutation (AP) (BF) (CM) (DS) (EW) (GY) (HK) (IU) (JX) (LZ) (NR) (OT).
Rice. 5. Zoo nkauj kho kom zoo nkauj, tab sis tsis haum rau encryption. Tsis tu ncua.
Txoj Hauj Lwm 1. Na fig. 4 koj muaj lwm qhov permutation los tsim Enigma. Luam daim duab rau xerograph. Tsim lub tsheb, sau koj lub npe thiab lub xeem. Kuv CWONUE JTRYGT. Yog tias koj xav khaws koj cov ntawv ntiag tug, siv Cardboard Enigma.
Txoj Hauj Lwm 2. Nkag mus rau koj lub npe thiab lub xeem lub npe ntawm ib qho ntawm "tsheb" koj pom, tab sis (tseem ceeb!) nrog rau qhov teeb meem ntxiv: peb tsis tig ib qho rau sab xis, tab sis raws li cov tswv yim {1, 2, 3, 2, 1, 2, 3, 2, 1, ....} - uas yog, ua ntej los ntawm ib tug, ces los ntawm ob, ces los ntawm peb, ces los ntawm 2, ces dua los ntawm 1, ces los ntawm 2, thiab lwm yam, xws li ib tug "wavelet" . Xyuas kom tseeb tias kuv lub npe thiab lub xeem raug encrypted li CZTTAK SDBITH. Tam sim no koj puas nkag siab tias lub tshuab Enigma muaj zog npaum li cas?
Kev daws teeb meem rau cov kawm tiav high school. Muaj pes tsawg configuration xaiv rau Enigma (nyob rau hauv no version, raws li tau piav nyob rau hauv tsab xov xwm)? Peb muaj 24 tsab ntawv. Peb xaiv thawj khub tsiaj ntawv - qhov no tuaj yeem ua tiav ntawm
txoj kev. Cov khub tom ntej tuaj yeem xaiv rau
txoj kev, ntau
lwm. Tom qab cov lej sib raug (tag nrho cov lej yuav tsum muab sib npaug), peb tau txais
151476660579404160000
Ces faib tus lej ntawd los ntawm 12! (12 factorial), vim tias tib khub tuaj yeem tau txais kev txiav txim sib txawv. Yog li thaum kawg peb tau txais "tag nrho"
316234143225,
uas tsuas yog ntau dua 300 billion, uas tsis zoo li tus lej loj heev rau niaj hnub supercomputers. Txawm li cas los xij, yog tias qhov kev txiav txim random ntawm lawv tus kheej raug coj mus rau hauv tus account, tus lej no nce ntau. Peb kuj tuaj yeem xav txog lwm hom kev hloov pauv.
Saib kuj:
