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NPTEL Computational Number Theory and Algebra Assignment 3 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

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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 f be a polynomial of degree n over C[x]C[x], where n is a power of 2. Let ωω be n-th primitive root of unity. Given (ω0,ω1,…,ωn−1)(ω0,ω1,…,ωn−1), what is the minimum number of CC operations in which you can compute (f(ω0),f(ω1),…,f(ωn−1))(f(ω0),f(ω1),…,f(ωn−1))?

O(n)O(n)

O(nlogn)O(nlog⁡n)

O(n2)O(n2)

O(n2logn)O(n2log⁡n)

Ans – C

1 point

Which of the following can be false for some n-th root of unity, ωω?

ω2ω2 is n/2-th root of unity, for even n.

ωn+k=ωkωn+k=ωk, for every integer k.

ωn/2+k=−ωkωn/2+k=−ωk, for every integer k and even n.

The n-th roots of unity form a cyclic group under multiplication.

Ans – D

1 point

Which of the following is false for polynomial multiplication of two degree n polynomials over R[x]?

If R has a n-th primitive root of unity and n is a power of 2, polynomial multiplication can be done in O(nlogn)O(nlog⁡n) R operations.

If R does not have a n-th primitive root of unity, polynomial multiplication can be done in O~(n)O~(n) R operations.

Polynomial multiplication can be done in O~(n)O~(n) R operations for any R.

None of these.

Ans – a

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1 point

Let f,g∈F[x]f,g∈F[x] be two polynomials, over a field FF, of degree at most ℓℓ. You learned in lectures about Schonhage-Strassen’s algorithm for fast multiplying ff with gg where ℓℓ was assumed to be a power of 2. In which of the following scenarios the algorithm does not work?

When the characteristic of FF is 00.

When the characteristic of FF is 2.

When the characteristic of FF is odd.

None of the above options.

Ans – A

1 point

In lectures, you saw the discrete fourier transform matrix DFT[ω]DFT[ω] where ωω is the primitive ℓℓth root of unity with ℓℓ, a power of 2. What is DFT[ω−1]⋅DFT[ω]DFT[ω−1]⋅DFT[ω] (IℓIℓ be the ℓ×ℓℓ×ℓ identity matrix)?

ℓ⋅Iℓℓ⋅Iℓ

IℓIℓ

ωℓ⋅Iℓωℓ⋅Iℓ

ℓ2⋅Iℓℓ2⋅Iℓ

Ans – C

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1 point

Let ZZ be the ring of integers and kk be a positive integer. In which of the following ring extensions of ZZ, the identity 1+y+y2+⋯+y2k−1=01+y+y2+⋯+y2k−1=0 holds?

Z[y]/⟨yk⟩Z[y]/⟨yk⟩

Z[y]/⟨yk−1⟩Z[y]/⟨yk−1⟩

Z[y]/⟨yk+1⟩Z[y]/⟨yk+1⟩

Z[y]/⟨y⟩

Ans – B

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