Question

Solve the below cryptography and probability questions:-


1. Poly-alphabetic Cipher

For this question, we consider the Hill cipher given in the textbook on an alphabet

A consisting of 26 English characters (A-Z), 10 numeric characters (0-9) and space,

which corresponds to integers 0 to 36. Here the plaintext is processed successively in

blocks of size m. The encryption algorithm takes a block with m plaintext digits and

transforms into a cipher block of size m using a key matrix of size mÅ~m by the linear

transformation, which is given by:

c1 = (k1,1p1 + k1,2p2 + . . . + k1,mpm) mod 37

c2 = (k2,1p1 + k2,2p2 + . . . + k2,mpm) mod 37

. . .

cm = (km,1p1 + km,2p2 + . . . + km,mpm) mod 37

Note: For this question, correspondence between plaintext and number modulo 37 are

as follows “A” ↔ 0, “B” ↔ 1, “C” ↔ 2, . . . , “Z” ↔ 25, “0” ↔ 26, “1” ↔ 27,

“2” ↔ 28, . . . , “9” ↔ 35 and “ ” (space) ↔ 36

(a) How many different keys are possible in this system?

(b) This cipher is easily broken with a known plaintext attack. An adversary discovers

the following ciphertext is encrypted using this cipher with m = 5 (55

characters in total, no spaces):

A8VS3XRDEON6JEVXGJID13C07L4C1R4Q965XWRA5DQGYWTNHYO4ND8Z

If the following combination of plaintext and ciphertext is given (please replace

both “?????” by the last five digits of your student number), decrypt the cipher

by giving the plaintext as well as both encryption and decryption keys.

Plaintext X9B6T6JAW3UEY7FHIW?????5Z

Ciphertext 2Q59ZZ1Z?????UMDNY2JHINTS

You need to show step-by-step details of your working. Make sure to include

the details of any package, functions used, and/or programs developed. Simply

showing the final result and/or a program would not receive marks.

2. Probability

Let x be the fourth digit of your ID eg.1123405 (without leading zero), y be the sixth

digit of your ID. The value N used in this task is given by 5x + 6y + 15.

For the below tasks, you need to show your working by providing formula used, and/or

short explanation. Also give the numerical final answer (e.g. 1024 instead of 210).

(a) What is your value of N based on your student ID? You may simply show N,

but please make sure that your calculation of N is correct, as you will need this

value for the rest of tasks.

(b) Assuming that we have 230 students enrolled in this subject, and all student

numbers are randomly generated. What’s the probability that at least one of

your classmate shares the same N with you? Your result should be rounded to

three digits after the decimal point.

(c) How many ways to place N different balls into five different bins?

(d) How many ways to place N identical balls into five different bins, so that all bins

are non-empty?

(e) How many ways to place N identical balls into five different bins?

(f) How many ways to place N identical balls into five identical bins, so that at most

two bins are non-empty?​

Answer 1

Answer:

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Explanation:

I need point so...