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COMP90043 Cryptography and Security
创建日期：2024-04-15 18:13:06
标签： COMP90043
Cryptography and Security

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COMP90043 Cryptography and Security

Assignment 1
Objectives
This assignment is designed to improve your understanding of the Euclid’s algorithm,
classical ciphers and basics of probability. It’s also aimed at improving your problem-solving
and written communication skills.
Questions
1. General Security [8 marks]
Which of the following factors might be the most concern by the public in regards to
the COVID passport?
Justify your answer in a few sentences.
(a) Confidentiality (b) Integrity (c) Availability
2. Classical Ciphers [20 marks]
Consider the following version of a classical cipher where plaintext and ciphertext
elements are the integers from 0 to 27. Note that this alphabet may be used when
the characters of plaintexts are elements the language A consisting of the 26 English
characters (A?Z) along with space. The encryption function, which maps any plaintext
p to a ciphertext c, is given by
c = E(a,b)(p) = (ap + b) mod 27,
where a and b are integers less than 27.
(a) For what values of a and b does a decryption function of E(a,b) exist? Write down
the decryption function.
(b) How many keys are possible for this scheme? Explain your reasoning.
(c) Would this cipher be considered as mono-alphabetic cipher or poly-alphabetic
cipher? Why?
(d) You are given a large amount of ciphertext characters encrypted using this scheme.
Assuming its plaintext was written in English, show how an attacker can retrieve
the key.
1
(e) An oracle is available to you which can output the encrypted ciphertext for
arbitrary plaintext you give. Briefly describe an efficient way to retrieve the
key using the oracle.
3. Euclid’s algorithm [15 marks]
Perform the following implementation tasks in a language of your choice. You are free
to employ any underlying integer arithmetic library. In order to get full marks, your
algorithm has to be able to work in realistic cryptographic environments (consider
101000 as input).
(a) Implement the extended GCD algorithm as discussed in lectures and print the
code here.
(b) Implement a function which takes two positive integers a, n as inputs, and returns
the inverse of (a mod n) based on your extended GCD algorithm (that you just
implemented above). Print the code for this function.
(c) Use the above function to find the inverse of X( mod 16811891), where X is your
student number. You don’t need to show steps for the calculation.
4. Poly-alphabetic Cipher [21 marks]
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 the following
special characters:
: ; < = [ (1)
which corresponds to integers 0 to 40. 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 m×m by the linear
transformation, which is given by:
c1 = (k1,1p1 + k1,2p2 + · · ·+ k1,mpm) mod 41
c2 = (k2,1p1 + k2,2p2 + · · ·+ k2,mpm) mod 41
· · ·
cm = (km,1p1 + km,2p2 + · · ·+ km,mpm) mod 41
Note: For this question, correspondence between plaintext and number modulo 41 are
as follows “A” ? 0, “B” ? 1, “C” ? 2, . . . , “Z” ? 25, “0”? 26, “1”? 27,
“2” ? 28, . . . , “9” ? 35, “ : ” ? 36, “; ” ? 37, “ < ” ? 38, “ = ” ? 39,
“[” ? 40
(a) The following is ciphertext where the encryption method described above has
been employed:
0ANJASCDOP7A4R82YQR[N11Z;AXCJNV9UO[06;;2U4;ZXWKW:V2BMV:9264:DGOPJSB=9L9:EF
2
where the key matrix is: ??????
29 1 5 0 26
28 17 38 25 8
37 40 1 26 14
40 33 31 34 14
31 23 29 12 23
?????? (2)
Find the corresponding plaintext.
(b) How many different keys are possible in this system described by the Question?
What if we disallowed the symbols “[”, “ = ”, “ < ” so as to only consider an
alphabet with 38 characters? In other words, now considering a Hill cipher
working mod38, how many possible keys are there?
(c) We return now to the full alphabet A. 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 XY[LER5HI;:LMOPQR?????SBJ
Ciphertext [WQ:9B0A?????CUQYou 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.
5. Polynomials [11 marks]
For this question, we work in the ring of polynomials with rational coefficients, Q[x] :=
{a0 + a1x + . . . + anxn | n ∈ N, n ≥ 0, ai ∈ Q}.
Definition 1. The greatest common divisor of two polynomials p, q ∈ Q[x] is a
polynomial f (unique up to multiplication by an invertible element of Q[x]) such that
f divides both p and q, and any common divisor of p and q also divides f .
(a) Find the greatest common divisor of the following polynomials:
f(x) := x4 ? 3x3 + 5x2 ? 17x + 6 and g(x) := x3 ? 4x2 + 4x + 6 (3)
Justify your answer.
(b) Describe (using pseudocode) an analogue of the Euclidean algorithm which finds
the greatest common divisor of two arbitrary polynomials.
3
(c) Perform your algorithm on the pair (3). You must show all steps of your algorithm.
(d) Find polynomials p, q ∈ Q[x] such that
f(x)p(x) + g(x)q(x) = gcd(f(x), g(x)) (4)
(e) What is the multiplicative inverse of g(x) mod x4 ? 3x3 ? 5x2 ? 17x + 6?
4
Submission and Evaluation
? You must submit a PDF document via the COMP90043 Assignment 1 submission
entry on the LMS by the due date. Handwritten, scanned images, and/or Microsoft
Word submissions are not acceptable — if you use Word, create a PDF version for
submission.
? Late submission will be possible, but a late submission will attract a penalty of 10%
per day (or part thereof). Requests for extensions on medical grounds will need to be
supported by a medical certificate. Any request received less than 48 hours before the
assessment date (or after the date) will generally not be accepted except in the most
extreme circumstances.
? This assignment will be marked out of 75 marks, and will contribute to 7.5% of your
total marks in this subject. Marks are primarily allocated for correctness of your
thinking and clarity of your communication, rather than (only) the correct result
without justification.
? We expect your work to be neat — parts of your submission that are difficult to read
or decipher will be deemed incorrect. Make sure that you have enough time towards
the end of the assignment to present your solutions carefully. Time you put in early
will usually turn out to be more productive than a last-minute effort.
? You are reminded that your submission for this assignment is to be your own individual
work. For many students, discussions with friends will form a natural part of the
undertaking of the assignment work. However, it is still an individual task. You
are welcome to discuss strategies to answer the questions, but not to share the work
(even draft solutions) on social media or discussion board. It is University policy
that cheating by students in any form is not permitted, and that work submitted for
assessment purposes must be the independent work of the student concerned. Answers
which are not original may not receive full marks even if they are correct.
Please see http://academicintegrity.unimelb.edu.au
If you have any questions, you are welcome to post them on the LMS discussion board
so long as you do not reveal details about your own solutions. You can also email the Head
Tutor, Will Troiani ([email protected]) or the Lecturer, Udaya Parampalli
([email protected]). In your message, make sure you include COMP90043 in the
subject header. In the body of your message, include a precise description of the problem.
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 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