# NPTEL Computational Number Theory and Algebra Assignment 4 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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*1 point*

LetÂ xxÂ be an element ofÂ Z/âšnâ©Z/âšnâ©. What is the time complexity (in bit operations) of computingÂ xdxdÂ inÂ Z/âšnâ©Z/âšnâ©?

O(d.n2)O(d.n2)

O(logd.n2)O(logâĄd.n2)

O(d.log2n)O(d.log2âĄn)

O(logd.log2n)O(logâĄd.log2âĄn)

Ans – A

*1 point*

What is the time complexity of multiplying twoÂ nĂnnĂnÂ lower triangular matrices?

O(n2)O(n2)Â usingÂ n/2Ăn/2n/2Ăn/2Â sized block multiplication.

O(nlog310)O(nlog3âĄ10)Â usingÂ n/3Ăn/3n/3Ăn/3Â sized block multiplication

Î(M(n))Î(M(n)), where M(n) is the time complexity of multiplying two generalÂ nĂnnĂnÂ matrices.

o(M(n))o(M(n)), where M(n) is the time complexity of multiplying two generalÂ nĂnnĂnÂ matrices.

Ans – B

*1 point*

Which of the following is false for matrix multiplication of twoÂ nĂnnĂnÂ matrices?

It can be done inÂ O(nlogn)O(nlogâĄn)Â time.

It can be done inÂ O(nlog27)O(nlog2âĄ7)Â time.

It can be done inÂ O(n3)O(n3)Â time.

It can be done inÂ O(n3logn)O(n3logâĄn)Â time.

Ans – C

*1 point*

Strassenâs recursive matrix multiplication algorithm reduces the number of multiplications over the naive method in each recursive step. For 2Ă22Ă2Â 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 Strassenâs method?

8 vs 6

7 vs 6

8 vs 7

7 vs 5

Ans – C

*1 point*

The naive algorithm for multiplying two nĂnnĂnÂ matrices requires O(n3)O(n3)Â multiplications. There has been a lot of work to improve this bound. What is the complexity of the current best algorithm?

O(n2.3728639)O(n2.3728639)

O(n2.3718639)O(n2.3718639)

O(n2.3729)O(n2.3729)

O(n2.376)O(n2.376)

Ans – B

*1 point*

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

- Order 33tensor rank computation is NP-hard.
- Rank of an order nntensor lies between n2n2and n3.n3.

Only 1.

Only 2.

Both 1 and 2.

Neither 1 nor 2.

Ans – D

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