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Unformatted text preview: MATH 135 Algebra, Solutions to Assignment 10 1: (a) The name of a mathematician was encoded using a Caesar cypher to give the cyphertext below. Decrypt the cyphertext. ZPXGDY Solution: We list all 26 possibilities. ZPXGDY GWENKF NDLURM TJRAXS AQYHEZ HXFOLG OEMVSN UKSBYT BRZIFA IYGPMH PFNWTO VLTCZU CSAJGB JZHQNI QGOXUP WMUDAV DTBKHC KAIROJ RHPYVQ XNVEBW EUCLID LBJSPK SIQZWR YOWFCX FVDMJE MCKTQL The only famous mathematician on the list is Euclid (the mathematician Kairoj is not as well known). (b) Two quotes from Bertrand Russell were encoded using a substitution cypher to give the cyphertext below. Use the fact that the most frequently occurring letters in English text are the letters E, T and A (in that order) to help decrypt the cyphertext. GMQEUBQI MWJMVI QMWS BZ XVUYL ZBE QOCUE PBDYQEV HDQ YCTCE BZ SUWWUYL ZBE QOCUE PBDYQEV. RMQOCRMQUPI RMV HC XCZUYCX MI QOC IDHACPQ UY JOUPO JC YCTCE SYBJ JOMQ JC MEC QMWSUYL MHBDQ, YBE JOCQOCE JOMQ JC MEC IMVUYL UI QEDC. Solution: By counting the number of occurrences of each letter, we find that C occurs most frequently, followed by Q and then by M, so we guess that we should make the substitutions C → E, Q → T and M → A. Then the coded word QOC in the second quote will become QOC → T–E so we guess that O → H so that QOC → THE (we could also have O → I or O → O, but O → H seems the most likely). The coded word JOMQ on the last line will then become JOMQ → –HAT, so we guess that J → W so that JOMQ → WHAT. Similarly, from JOCQOCE → WHETHE–, we guess that E → R , then from QOCUE → THE–R, we guess that U → I, then from QEDC → TR–E, we guess that D → U, then from JOUPO → WHI–H, we guess that P → C, from RMQOCRMQUPI → –ATHE–ATIC–, we guess that R → M and I → S. After a few more such guesses, we find that the first quote becomes “Patriots always talk of dying for their country but never of killing for their country”, and the second quote becomes “Mathematics may be defined as the subject in which we never know...
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 Fall '08
 ANDREWCHILDS
 Algebra, Cryptography, Prime number, Bertrand Russell, 3rd millennium

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