Cryptography Reference
In-Depth Information
n
=
103263461981882861062786663752777462175094289042793826073777660932064239
421442774128148179956149383630351472101383901416222437739428120537496844
245199079442668017154702976062842089841568938071479001427994543188554221
115276088728567455684565042964557431421107222436249019878609266108175858
124361197755381114937973572181548712613409747227913581319056985394852398
084487124970716064947426472124245378223284977468290053203588152259348515
625046935987244280800174540272797917758248171254341207690298062481216167
528333934836121894789925388081585752480608066799998740549915224135659234
5912472109440411430331086757929071611556082,
and
(
n
*)
=
565665541752874160808995452716541646884589640079627004022468110947317915
729355650576328262639716563312562271639231787408458611699081599251982353
481167184847828229145234319872739621266282060173792951287146944485307550
598063408075929912008320329707348918335177023197781310971354341674303782
991390796732216220068702660726645223525825601204462280782794956836356123
348142830696396398621571457726481064542977798874551230913235978372189710
059519841017844537104921977286499738353103100370267565323698898530108734
170588312885569924743018308407355426370867568374322691255471122749516099
67399232498190089468651608956593592819968.
He then combines these values using the CRT formula to obtain
2
m
x
107297411861321680907514861384339217765295082808206114325123089381136015
524701635341161871377007555148580336419555532871152609983017836612563487
591327437743663832845605641026043394022012578434187301314894547456727730
257625586451817991411251446477064513852860540591300380302198128398350095
008517337820821682833970271741889770829366774041010862939865154136215583
576547154338003222682025233175666362486458726928146033065226496988099344
946501245683372299902380475067363053365780390691996437208014464278584234
314944962513348436839550335887834620312109611730644090035255616166589958
290436903597929892282050969055549903864227200176485400546369403199956019
817111464120200227052607334143654876555969639885312017068888815665743235
084504551643813143206463824739078205795097842594282570823541551058657452
56509992538572757424398951566596819756096784 (mod
n
n
*).
Since he knows
2
is less than
2
(rather than some residue
m
n
n
*,
x
must in fact equal
m
2
). He merely needs to compute the positive square root of
of
m
x
to regain the plaintext:
m
=
√
x
=
327562836508236509237590237590823750923875098275908237590827359082375908
723095873209875093285790328750932875093248750983275098327509832759082375
098370957309287509328750923858723658972365892365930275094327590342857326
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