E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 11: Caesar cypher1. ObjectiveIn this worksheet we crack the Caesar cypher using statistical analysis.2. Statistical analysisLetter PercentageE11.16A8.50R7.58I7.54O7.16T6.95N6.65S5.74L5.49C4.54U3.63D3.38P3.17Letter PercentageM3.01H3.00G2.47B2.07F1.81Y1.78W1.29K1.10V1.01X0.29Z0.27J0.20Q0.20The frequency of letters is relevant for designing keyboards. The Kwerty keyboard for examplehas ESER and OI in prominent places.3. More tricksThe ’top twelve’ letters help with about 80 percent of the text. You can remember the first 8 withthe memnonic”A SIN TO ERR”.An other thing to look for: The top pairs which appear areTH HE AN RE ER IN ON AT ND ST ES EN OF TE ED OR TI HI AS TOThe most frequent double letters are”LL EE SS OO TT FF RR NN PP CC”4. An exampleWe aim to decrypt the following text:xf uif qfpqmf pg uif vojufe tubuft,jo psefs up gpsn b npsf qfsgfdu vojpo,ftubcmjti kvtujdf, jotvsf epnftujd usborvjmjuz,qspwjef gps uif dpnnpo efgfodf,qspnpuf uif hfofsbm xfmgbsf,boe tfdvsf uif cmfttjoht pg mjcfsuzup pvstfmwft boe pvs qptufsjuz,ep psebjo boe ftubcmjti uijt dpotujuvujpo gps uifvojufe tubuft pg bnfsjdb5. Decoding1) We count the number of letters which occur. Since we have not much time, the 8 most frequentletters are listed in this text. Can you figure out the text?f appears 39 timesu appears 29 timesp appears 25 timest appears 20 timess appears 20 timesj appears 20 timeso appears 17 timesb appears 14 timesE-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 11: The Vigen`ere Cipher1. ObjectiveWe learn how to encrypt messages using the Vigen`ere Cipher. This encryption was used for along time and should be seen as an important marker in the development of substitution ciphers:Julius Caesar -70Ahmad al-Qalqashandi 1400Leon Battista Alberti 1467Johannes Trithemius 1508Blaise de Vigin`ere 1586Charles Babbage 1854Friedrich Kasiski 1863Arthur Scherbius 1920Blaise de Vigen`ere2. An exampleAssume we have a secret key like ”ENIGMA”. Given a text like ”HARVARD IS COOL”, weencrypt it using the following table: for the first letter, we use the line starting with E, for thesecond letter, we use the line starting with N etc.A B C D E F G H I J K L M N O P Q R S T U V W X Y ZA A B C D E F G H I J K L M N O P Q R S T U V W X Y ZB B C D E F G H I J K L M N O P Q R S T U V W X Y Z AC C D E F G H I J K L M N O P Q R S T U V W X Y Z A BD D E F G H I J K L M N O P Q R S T U V W X Y Z A B CE E F G H I J K L M N O P Q R S T U V W X Y Z A B C DF F G H I J K L M N O P Q R S T U V W X Y Z A B C D EG G H I J K L M N O P Q R S T U V W X Y Z A B C D E FH H I J K L M N O P Q R S T U V W X Y Z A B C D E F GI I J K L M N O P Q R S T U V W X Y Z A B C D E F G HJ J K L M N O P Q R S T U V W X Y Z A B C D E F G H IK K L M N O P Q R S T U V W X Y Z A B C D E F G H I JL L M N O P Q R S T U V W X Y Z A B C D E F G H I J KM M N O P Q R S T U V W X Y Z A B C D E F G H I J K LN N O P Q R S T U V W X Y Z A B C D E F G H I J K L MO O P Q R S T U V W X Y Z A B C D E F G H I J K L M NP P Q R S T U V W X Y Z A B C D E F G H I J K L M N OQ Q R S T U V W X Y Z A B C D E F G H I J K L M N O PR R S T U V W X Y Z A B C D E F G H I J K L M N O P QS S T U V W X Y Z A B C D E F G H I J K L M N O P Q RT T U V W X Y Z A B C D E F G H I J K L M N O P Q R SU U V W X Y Z A B C D E F G H I J K L M N O P Q R S TV V W X Y Z A B C D E F G H I J K L M N O P Q R S T UW W X Y Z A B C D E F G H I J K L M N O P Q R S T U VX X Y Z A B C D E F G H I J K L M N O P Q R S T U V WY Y Z A B C D E F G H I J K L M N O P Q R S T U V W XZ Z A B C D E F G H I J K L M N O P Q R S T U V W X Y3. Do it yourselfENIGMAE NI GMAEHARVARD IS COOLE-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 11: RSA Encryption1. ObjectiveWe want to understand the basic mechanism for RSA encryption.Ron Rivest, Adi Shamir and Len Adleman.2. The methodAn RSA public key is a pair (n, a) where n is an integer with secret factorization n = pq andwhere a < (p − 1)(q − 1) is such that there exists b with ab = 1 mod (p − 1)(q − 1). Ana publishesthis pair. If Bob wants to send a secrete message to Ana, he transmits to Ana the messagey = xamod n .Ana can read the email by computingybmod n .3. Why does it work?We use the Fermat’s little theorem which tells that xp−1− 1 is is divisible by p and xq−1− 1 isdivisible by q. But this assumes p, q to be prime. For n = pq, we have x(p−1)(q−1)− 1 divisible by pq.Problem 1) Take p = 3 and q = 5 Verify that 2(p−1)(q−1)− 1 is divisible by …