# NPTEL Computational Number Theory and Algebra Assignment 5 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**

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*BELOW YOU CAN GET YOUR NPTEL Computational Number Theory and Algebra Assignment 5 Answer 2022***?** :

**?**:

*1 point*

x2+1×2+1Â is reducible over which of the following rings?

F2[x]F2[x]

F5[x]F5[x]

GF(4)[x]GF(4)[x]

All of these

Ans – C

*1 point*

LetÂ FFÂ be a finite field of sizeÂ qqÂ andÂ EEÂ be some finite field extension ofÂ FF. Then, which of the following is true?

(a+b)q=aq+bq(a+b)q=aq+bqÂ inÂ EE, for allÂ a,bâEa,bâE.

(a+b)q=aq+bq(a+b)q=aq+bqÂ inÂ FF, for allÂ a,bâFa,bâF.

(a+b)q=a+b(a+b)q=a+bÂ inÂ FF, for allÂ a,bâFa,bâF.

All of these.

Ans –Â A

*1 point*

Suppose there is a factorization algorithm inÂ F[x]F[x]Â that computes only a single factor of the input polynomial (if it exists) inÂ poly(d,logq)poly(d,logâĄq)Â time, where d is the degree of given polynomial and q is the size of finite field F. Then, which of the following is true?

There is an algorithm for irreducibility testing of polynomials inÂ F[x]F[x]Â which takesÂ poly(d,logq)poly(d,logâĄq)Â time.

There is a factorisation algorithm which can find all the factors of a given polynomial overÂ F[x]F[x]Â and takes onlyÂ poly(d,logq)poly(d,logâĄq)Â time.

Both of these.

None of these.

Ans –Â B

*1 point*

Let FqFq be a finite field. Which of the following is true?

FâqFqâÂ Â is a cyclic group of order q.

FâqFqâÂ is a cyclic group of order q-1.

FâqFqâÂ is not a cyclic group.

None of the above.

Ans –Â A

*1 point*

Let ff be a non-zero polynomial in Fq[x]Fq[x] where FqFq is a finite field of characteristic pp. If ââx(f)=0ââx(f)=0 then which of the following is necessarily true?

- ff is a constant in FqFq. 2. There exists hh in Fq[x]Fq[x] such that f=hpf=hp.

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

Ans –Â B

*1 point*

Let ff be a square-free polynomial in Fq[x]Fq[x] where FqFq is a finite field. If we can find two polynomials gg and hh in Fq[x]Fq[x] of degrees less than the degree of ff such that gâ hâĄ0modfgâ hâĄ0modf, then which of the following is necessarily true?

- gg and hh are factors of ff.
- GCD of gg and hh with ff gives non-trivial factors of ff.

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

Ans –Â B

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