NPTEL Computational Number Theory and Algebra Assignment 5 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

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

$x^{2} + 1$ is reducible over which of the following rings?

$F_{2} [x]$

$F_{5} [x]$

$G F (4) [x]$

All of these

Ans – C

1 point

Let $F$ be a finite field of size $q$ and $E$ be some finite field extension of $F$ . Then, which of the following is true?

$(a + b)^{q} = a^{q} + b^{q}$ in $E$ , for all $a, b \in E$ .

$(a + b)^{q} = a^{q} + b^{q}$ in $F$ , for all $a, b \in F$ .

$(a + b)^{q} = a + b$ in $F$ , for all $a, b \in F$ .

All of these.

Ans – A

1 point

Suppose there is a factorization algorithm in $F [x]$ that computes only a single factor of the input polynomial (if it exists) in $p o l y (d, \log q)$ 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 in $F [x]$ which takes $p o l y (d, \log q)$ time.

There is a factorisation algorithm which can find all the factors of a given polynomial over $F [x]$ and takes only $p o l y (d, \log q)$ time.

Both of these.

None of these.

Ans – B

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1 point

Let $F_{q}$ be a finite field. Which of the following is true?

$F_{q}^{*}$ is a cyclic group of order q.

$F_{q}^{*}$ is a cyclic group of order q-1.

$F_{q}^{*}$ is not a cyclic group.

None of the above.

Ans – A

1 point

Let $f$ be a non-zero polynomial in $F_{q} [x]$ where $F_{q}$ is a finite field of characteristic $p$ . If $\frac{\partial}{\partial x} (f) = 0$ then which of the following is necessarily true?

$f$ is a constant in $F_{q}$ . 2. There exists $h$ in $F_{q} [x]$ such that $f = h^{p}$ .

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

Ans – B

1 point

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Let $f$ be a square-free polynomial in $F_{q} [x]$ where $F_{q}$ is a finite field. If we can find two polynomials $g$ and $h$ in $F_{q} [x]$ of degrees less than the degree of $f$ such that $g \cdot h \equiv 0 mod f$ , then which of the following is necessarily true?

$g$ and $h$ are factors of $f$ .
GCD of $g$ and $h$ with $f$ gives non-trivial factors of $f$ .

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

Ans – B

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NPTEL Computational Number Theory and Algebra Assignment

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