NPTEL Computational Number Theory and Algebra Assignment 4 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

Let $x$ be an element of $Z / ⟨ n ⟩$ . What is the time complexity (in bit operations) of computing $x^{d}$ in $Z / ⟨ n ⟩$ ?

$O (d . n^{2})$

$O (\log d . n^{2})$

$O (d . \log^{2} n)$

$O (\log d . \log^{2} n)$

Ans – A

1 point

What is the time complexity of multiplying two $n \times n$ lower triangular matrices?

$O (n^{2})$ using $n / 2 \times n / 2$ sized block multiplication.

$O (n^{\log_{3} 10})$ using $n / 3 \times n / 3$ sized block multiplication

$Θ (M (n))$ , where M(n) is the time complexity of multiplying two general $n \times n$ matrices.

$o (M (n))$ , where M(n) is the time complexity of multiplying two general $n \times n$ matrices.

Ans – B

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1 point

Which of the following is false for matrix multiplication of two $n \times n$ matrices?

It can be done in $O (n \log n)$ time.

It can be done in $O (n^{\log_{2} 7})$ time.

It can be done in $O (n^{3})$ time.

It can be done in $O (n^{3} \log n)$ time.

Ans – C

1 point

Strassen’s recursive matrix multiplication algorithm reduces the number of multiplications over the naive method in each recursive step. For $2 \times 2$ matrices A and B, what is the number of multiplications required by the naive method (multiplying each row of A with each column of B) vs Strassen’s method?

8 vs 6

7 vs 6

8 vs 7

7 vs 5

Ans – C

1 point

The naive algorithm for multiplying two $n \times n$ matrices requires $O (n^{3})$ multiplications. There has been a lot of work to improve this bound. What is the complexity of the current best algorithm?

$O (n^{2.3728639})$

$O (n^{2.3718639})$

$O (n^{2.3729})$

$O (n^{2.376})$

Ans – B

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1 point

Revise the definition and facts about Tensors from lectures and answer which of the following is correct?

Order $3$ tensor rank computation is NP-hard.
Rank of an order $n$ tensor lies between $n^{2}$ and $n^{3} .$

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

Ans – D

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NPTEL Computational Number Theory and Algebra Assignment

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