# NPTEL Computational Number Theory and Algebra Assignment 2 Answer

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## NPTEL Computational Number Theory and Algebra Assignment

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 AUDIENCEÂ :Â**Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.

**Next Week Assignment Answers**

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.

- 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 â quizzes and programming assignments).Â**- Unproctored programming exam score = 25% of the average scores obtained as part of Unproctored programming exam â 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.Â**

**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.Â**

**CHECK HERE OTHERS NPTEL ASSIGNMENTS ANSWERSÂ **

*BELOW YOU CAN GET YOUR NPTEL Computational Number Theory and Algebra Assignment 2 Answer 2022***?** :

**?**:

*1 point*

Given two integersÂ aa and bbÂ with |a|>|b||a|>|b|. In lectures, you saw that in each **round** Euclidâs gcd algorithm computes two integers cc and ddÂ 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 (ânâânâÂ denotes the least integer greater than or equal to nn)?

âlogaââlogâĄaâ

âlogbââlogâĄbâ

âlogaâ+âlogbââlogâĄaâ+âlogâĄbâ

âlogaââ âlogbââlogâĄaââ âlogâĄbâ

ans –Â b

*1 point*

Euclidâs 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 Euclidâs gcd algorithm?

- Computing aâ1modbaâ1modb.
- Computing the parameters in Bezoutâs identity for aa and bb.

Only 1.

Only 2.

Both 1 and 2.

None of 1 and 2.

ans –Â 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 âša,bâ©âša,bâ© of ring of integers ZZ by adding each element of ideal âšaâ©âšaâ© to each element of ideal âšbâ©âšbâ©, formally written as âša,bâ©={ca+db|âc,dâZ}âša,bâ©={ca+db|âc,dâZ}.

Let a=14a=14 and b=21b=21, which of the following ideals is same as the ideal âš14,21â©âš14,21â©?

âš2â©âš2â©

âš3â©âš3â©

âš14,28â©âš14,28â©

âš7â©âš7â©

ans –Â c

*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/âš36â©â Z/âš2â©ĂZ/âš18â©Z/âš36â©â Z/âš2â©ĂZ/âš18â©

Z/âš36â©â Z/âš3â©ĂZ/âš12â©Z/âš36â©â Z/âš3â©ĂZ/âš12â©

Z/âš36â©â Z/âš4â©ĂZ/âš9â©Z/âš36â©â Z/âš4â©ĂZ/âš9â©

Z/âš36â©â Z/âš6â©ĂZ/âš6â©Z/âš36â©â Z/âš6â©ĂZ/âš6â©

ans –Â b

*1 point*

ComputeÂ 7â1(modÂ 5)7â1(modÂ 5)Â andÂ 5â1(modÂ 7)5â1(modÂ 7)Â respectively.

3, 3

3, 5

2, 2

4, 3

ans –Â b

*1 point*

LetÂ x=3(modÂ 5)x=3(modÂ 5)Â andÂ x=2(modÂ 7)x=2(modÂ 7). Using CRT, computeÂ x(modÂ 35)x(modÂ 35).

13

23

30

33

ans –Â b

*1 point*

LetÂ x=3(modÂ 5)x=3(modÂ 5)Â andÂ x=2(modÂ 7)x=2(modÂ 7). ComputeÂ x(1023+2)(modÂ 35)x(1023+2)(modÂ 35). Hint: Use power of CRT to simplify calculations.

23

29

17

0

ans –Â 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(n2logâĄn)

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

O(n),O(n2)

ans –Â b

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