# NPTEL Computational Number Theory and Algebra Assignment 7 Answer

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**â€śNPTEL Computational Number Theory and Algebra Assignment 7 Answerâ€ť**

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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**

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Assignment not submitted

*1 point*

Consider unique decoding Reed Solomon code, RS:Â (Fq)kâ†’(Fq)n(Fq)kâ†’(Fq)n, which views a k-length message as a polynomial of degree k-1 inÂ Fq[x]Fq[x]. Given any two codewords, in at least how many positions must they differ? (distance of code)

n/k

k/n

n-k+1

n-k-1

AnsÂ – D

*1 point*

Alice uses Reed Solomon code, RS:Â (F7)3â†’(F7)5(F7)3â†’(F7)5Â to communicate with Bob. For message (2,0,1), Alice sends codeword (1,3,2,5,5) to Bob and for message (0,1,4), Alice sends codeword (4,5,6,0,1) to Bob. Can you guess the codeword which Alice will send for the message (2,1,5)?

(4,5,3,5,4)

(5,1,1,5,6)

(3,2,4,2,3)

We cannot guess due to insufficient information

AnsÂ – C

*1 point*

Consider Reed Solomon code, RS:Â (F3)2â†’(F3)3(F3)2â†’(F3)3, where a message (a,b) is viewed as polynomial ax+b inÂ F3[x]F3[x]Â and its evaluations are sent as codewords. Suppose channel introduces at most one error. Consider the list decoding version which outputs all valid codewords that agree on at least 2 places. Which of the following is true?

For received code (0,0,1), list decoding outputs { (0,0,0), (2,0,1), (0,2,1) }.

For received code (1,1,0), list decoding outputs { (1,1,1), (2,1,0), (0,1,0) }.

Both of these.

None of these.

AnsÂ – C

*1 point*

How many degree â„“â„“ polynomials are there in the finite field FpFp (p>2p>2 is prime)? How many of them are irreducible polynomials as shown in lectures?

pâ„“pâ„“ polynomials of degree â„“â„“ and at least pâ„“/2â„“pâ„“/2â„“ irreducible polynomials.

â„“â‹…pâ„“â„“â‹…pâ„“ polynomials of degree â„“â„“ and exactly pâ„“pâ„“ irreducible polynomials.

2pâ„“2pâ„“ polynomials of degree â„“â„“ and exactly pâ„“pâ„“ irreducible polynomials.

None of the above is correct.

AnsÂ – B

*1 point*

LetÂ f(x,y)f(x,y)Â be a square-free bivariate polynomial over the finite fieldÂ FpFp,Â p>2p>2Â is prime. IfÂ fmodyfmodyÂ is square-full, then which of the following is true?

There is a shift f to f(x+t,y), for some t>0, such that f mod y is square-free.

There is a shift f to f(x,y+t), for some t>0, such that f mod y is square-free.

There are shifts with respect to both x and y such that f mod y becomes square-free.

There may not be any shift at all which makes f mod y square-free.

AnsÂ – B

*1 point*

Which of the following is true about Kaltofen’s bivariate factoring algorithm over finite fields (as shown in lectures)?

1. It reduces bivariate factoring to univariate factoring efficiently.

2. It uses Hensel’s lifting to achieve the above reduction.

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

AnsÂ – D

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