Cryptography Reference
In-Depth Information
880797251065757743055215064989964046890133812129409097921942823451284700
353341417572617870433870197661339699702384692398942940091526767537122407
294622549228222879740566332018250812637451109162228995890265099106862175
129783360538190048124955153727405933205488194213497935668384446131585719
769748108125895563260802289655241774630887967226547180792062432701705774
9253168131337219010124364276404953144761357.
To compute the ciphertext
C
m
2
(mod
n
) for the first entity, we obtain
C
=
633117525812174963141726511411569613510692954921413433904834216758854296
633124876624725803389226024659872532640785232933916621224271521661591779
805056806132874825319189837524810853175640040181499810175228697753511769
733199644184786773309701377090720855844334771741032864292654831554262834
830106099159752454122074338825214233151941406426968586422774039868803500
958440877983431882318911204475101540253926424708618519209984525553968321
269537413569633044293116969006410634311794364957784418800457585030758560
064753995190942293115578955198138212298271399040518748965782046565242764
02217687340373577434217196605168494458628,
2
(mod
and we apply the transformation
C
*
m
n
*) to obtain the ciphertext for the second
entity:
C
* =
394135248113853837239995785378155119660436161585799942724612487297884404
632745939484613684148169252487434528179060956986090118025870479973165415
807243287982752578261206149474414115340822662855315835762700720048234558
712444543617330299373057364169102752460252141023980586511691405908268083
588138459388456857535076380649491418151420595527403167925459720523465715
021500179306273478205638799251490159856470610252397459099946884001280830
252629399799543352087268619877293435351018907962347103035620025259332348
275482523386757262089321203787452631835007817304852012889299139015826664
7501363641100908149923744524224707388555291.
An eavesdropper captures both messages
C
and
C
*, and knows their plaintext equivalents
are the same. He has access to the public moduli
n
, and
n
*, so he simply needs to find the
simultaneous solution
x
to
x
C
(mod
n
)
x
C
n
*).
CRT tells him exactly how to compute this solution; it is
* (mod
x
Cn
*(
n
*)
+
C
*
nn
(mod
n
n
*)
where (
*. Both of these
values are easily computable using the extended Euclidean algorithm. The values he obtains
for
n
*)
is an inverse of
n
* modulo
n
, and
n
is an inverse of
n
modulo
n
n
and (
n
*)
are
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