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Questions  |  score: 7  |  Solved By 139 People  |  35246 views  |  since Apr 07, 2008 - 01:22:52

Save the World (Crypto)

It is year 2018, the world war III is upcoming between USA and China. You
are a secret agent working for the USA. The USA IGA (Information Gathering
Agency) gathered three RSA enciphered messages. All messages were originated
from the same "clear text message", which is a symmetric key (m). With this
symmetric key there is a document encrypted. This document contains all the
secret information about the newest high-tech weapons made in China. This
document was symmetrically encrypted with the CSEA (Chinese Super Encryption
Algorythm). This algorythm is almost unbreakable.

You know, that the chinese government used RSA to encrypt the symmetric key.
Your boss, general Eromzig bought a new supercomputer to factorize
the RSA public keys. Scientists say they need 2 weeks to factorize one of
the public keys.
That's why he hired you, because he thinks you can break the RSA algorithm in
1 hour. Your mission is to get the symmetric key in decimal format.

The public keys which were used to encrypt the symmetric key are the following:

1. e1=3

n1=6710833728513015284114255704897983639265932189185710122673233728187031921
9630063308425261001094568350046449570317350200345069218868069937257334549833
4488724766148765407843262238743292578704036171018334438216003870719162685398
0909371868024916680863506138780750868544480874747891485377406501549284653034
6393353
It is the public key of general Sun Tzu.

The encrypted message sent to general Sun Tzu was:

c1=5905834709610937116879525212284626416445185838650537322471725138676007071
5998321616322872045496360926010965597694014530236569746798289883620698223974
5953679160857179783215294239730426141919154687073223320914550743456925735158
5794030657232714040035046280998403417415439969323916725270200415847561582291
3969287

where c1= m^e1 mod n1

2. e2=3

n2=7229182064480185105033084811015940920204891953452795516412030276239664458
7973560056961559507263386276590810913644140431966430239440589187607901445455
5064768446425517222613332761974284232596856817882979785065536406882469923774
4054358642264643142596395741955735422343101742596135638430962118605087100390
8824523
It is the public key of general Qin Jiushao.

The encrypted message sent to general Qin Jiushao was:

c2=6883263686330952184273822891213591005924705168937975803056469379223409544
6146808098880880327706725969407906645386006921749510378088144330242695605459
5239198472987796880788709117604078885079308425746153831594914309208073700240
4840007500340125359757260591604068743351078048379311774951289024843307735044
171207

where c2= m^e2 mod n2

3. e3=3

n3=1047837889878270679300274527866088293820691272220456605149682021292026284
1945625442842048238092899277498433289776876178998918930600743208387416550053
3674664850382891750853847082861247537097054898628900960779958023749459758924
8806794509169565536014318238391241867494598197275008237041687608289913221668
78406679
It is the public key of general Ymenetn Gnauq.

The encrypted message sent to general Ymenetn Gnauq was:

c3=3153023395443523487161547446108225786081457485251683234856675074524798052
9778060741121062564921306275042716802719690563904928683994299563976201039979
0064561806373678683494380463451947999040218626967098053850845939291619220923
2800445091886965624967182389068673275945119737166044769624649368107209246409
2210096

where c3= m^e3 mod n3

You are lucky, because you have everything to decrypt the symmetric key (m):
You got brain, you have access to all knowledge in the world (wikipedia),
you got pencil and paper, and of course a "BNC" (Big Number Calculator).
When you break the code in 1 hour, you can save the world, because the
chinese will not attack you, if you have their secret weapon. So better
hurry. The World needs YOU!

The solution is the last 20 digits of the symmetric key(m)

Your solution for Save the World
Answer
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