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NPTEL Computational Number Theory and Algebra Assignment 2 Answer

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NPTEL Computational Number Theory and Algebra Assignment

ABOUT THE COURSE :
Algebra plays an important role in both finding algorithms, and understanding the limitations of computation. This course will focus on some of the fundamental algebraic concepts that arise in computation, and the algebraic algorithms that have applications in real life. The course will cover the problems of fast integer (or polynomial) multiplication (or factoring), fast matrix multiplication, primality testing, computing discrete logarithm, error-correcting codes, lattice- based cryptography, etc. The course intends to introduce both basic concepts and practical applications.

INTENDED AUDIENCE  : Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.
PREREQUISITES  : Preferable (but not necessary)– Theory of Computation, Algorithms, Algebra
INDUSTRIES  SUPPORT     : Cryptography, Coding theory, Computer Algebra, Symbolic Computing Software, Cyber Security, Learning Software

Next Week Assignment Answers

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This course can have Associate in Nursing unproctored programming communication conjointly excluding the Proctored communication, please check announcement section for date and time. The programming communication can have a weightage of twenty fifth towards the ultimate score.

Final score = Assignment score + Unproctored programming exam score + Proctored Exam score
  • Assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
  • ( All assignments in a particular week will be counted towards final scoring – quizzes and programming assignments). 
  • Unproctored programming exam score = 25% of the average scores obtained as part of Unproctored programming exam – out of 100
  • Proctored Exam score =50% of the proctored certification exam score out of 100
YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF ASSIGNMENT SCORE >=10/25 AND
UNPROCTORED PROGRAMMING EXAM SCORE >=10/25 AND PROCTORED EXAM SCORE >= 20/50. 
If any one of the 3 criteria is not met, you will not be eligible for the certificate even if the Final score >= 40/100. 

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1 point

Given two integers  aa and bb with |a|>|b||a|>|b|. In lectures, you saw that in each round Euclid’s gcd algorithm computes two integers cc and dd such that gcd(a,b)=gcd(c,d)gcd(a,b)=gcd(c,d).

Which of the following gives the best estimate on the number of rounds the algorithm takes (⌈n⌉⌈n⌉ denotes the least integer greater than or equal to nn)?

⌈loga⌉⌈log⁡a⌉

⌈logb⌉⌈log⁡b⌉

⌈loga⌉+⌈logb⌉⌈log⁡a⌉+⌈log⁡b⌉

⌈loga⌉⋅⌈logb⌉⌈log⁡a⌉⋅⌈log⁡b⌉

ans –  b

1 point

Euclid’s gcd algorithm is a very efficient algorithm with lots of applications. Let aa and bb are given co-prime integers. Which of the following problems can be solved efficiently using Euclid’s gcd algorithm?

  1. Computing a−1modba−1modb.
  2. Computing the parameters in Bezout’s identity for aa and bb.

Only 1.

Only 2.

Both 1 and 2.

None of 1 and 2.

ans –  b

1 point

You have seen the definition of an ideal of a ring. Given two integers aa and bb, you can generate an ideal ⟨a,b⟩⟨a,b⟩ of ring of integers ZZ by adding each element of ideal ⟨a⟩⟨a⟩ to each element of ideal ⟨b⟩⟨b⟩, formally written as ⟨a,b⟩={ca+db|∀c,d∈Z}⟨a,b⟩={ca+db|∀c,d∈Z}.

Let a=14a=14 and b=21b=21, which of the following ideals is same as the ideal ⟨14,21⟩⟨14,21⟩?

⟨2⟩⟨2⟩

⟨3⟩⟨3⟩

⟨14,28⟩⟨14,28⟩

⟨7⟩⟨7⟩

ans –  c

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1 point

In lectures, you learned about Chinese Remainder Theorem (CRT). Which of the following can be deduced by CRT (ZZ be the ring of integers)?

Z/⟨36⟩≅Z/⟨2⟩×Z/⟨18⟩Z/⟨36⟩≅Z/⟨2⟩×Z/⟨18⟩

Z/⟨36⟩≅Z/⟨3⟩×Z/⟨12⟩Z/⟨36⟩≅Z/⟨3⟩×Z/⟨12⟩

Z/⟨36⟩≅Z/⟨4⟩×Z/⟨9⟩Z/⟨36⟩≅Z/⟨4⟩×Z/⟨9⟩

Z/⟨36⟩≅Z/⟨6⟩×Z/⟨6⟩Z/⟨36⟩≅Z/⟨6⟩×Z/⟨6⟩

ans –  b

1 point

Compute 7−1(mod 5)7−1(mod 5) and 5−1(mod 7)5−1(mod 7) respectively.

3, 3

3, 5

2, 2

4, 3

ans –  b

1 point

Let x=3(mod 5)x=3(mod 5) and x=2(mod 7)x=2(mod 7). Using CRT, compute x(mod 35)x(mod 35).

13

23

30

33

ans –  b

1 point

Let x=3(mod 5)x=3(mod 5) and x=2(mod 7)x=2(mod 7). Compute x(1023+2)(mod 35)x(1023+2)(mod 35). Hint: Use power of CRT to simplify calculations.

23

29

17

0

ans –  a

1 point

Find an efficient algorithm for evaluating a polynomial. In how many R (ring) operations can you evaluate a polynomial of degree n on a single point from R using this algorithm? How much time does it take to evaluate the polynomial on n+1 points using this algorithm n+1 times? In the options (x,y) below – x should be the answer of first question and y should be the answer of second question. As a thinking exercise, compare this with the DFT step of fast multiplication algorithm done in class.

O(n2),O(n3)O(n2),O(n3)

O(n2),O(n2logn)O(n2),O(n2log⁡n)

O(1),O(n)O(1),O(n)

O(n),O(n2)

ans –  b

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