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WSN 123 (2019) 102-113 EISSN 2392-2192

Quantum of Integers 21 and 91 using Shor’s

Poornima Ardhyamath1, N. M. Naghabhushana2, Rohitha Ujjinimatad3 1Research Scholar, Department of Physics, RYM Engineering College, Ballari, India E-mail address: [email protected] 2Department of Physics, RYM Engineering College, Ballari, India E-mail address: [email protected] 3Department of Electronics, Proudhadevaraya Institute of Technology, Hosapete, India E-mail address: [email protected]

ABSTRACT In this paper we focused on the factorization of integer in detail using well known Shor’s algorithm and its quantum part realization. The algorithm finds prime factors any integer more efficiently than any known classical algorithm. It is based on prior knowledge of the answer to the factorization problem. Algorithm includes finding gcd using polynomial time , determination of unknown period from quantum computer and continued fraction expansion approach. Factorization of two integers 21 and 91 are shown in this paper with all steps. Quantum part of the factorization described mathematically.

Keywords: Shor’s Algorithm, Quantum Fourier Transform, GCD, Euclidean Algorithm, Factorization

1. INTRODUCTION

Shor’s Algorithm was named after its creator , who first published the algorithm in 1994. Quantum computers had already been discussed for many years by researcher and popular media, but it with Shor’s discovery that the potential of quantum

( Received 12 February 2019; Accepted 02 March 2019; Date of Publication 03 March 2019 ) World Scientific News 123 (2019) 102-113

computers was beginning to be understood. Shor’s algorithm has awoken a curiosity among physicists, mathematicians and computer scientists alike. After all, could other problems that are classically very difficult, or require an excessive amount of resources. Researchers say that these problems can be easily solvable on a quantum computer. The factorization of large numbers on classical computers is extremely resource demanding. Theoretical concepts of shor’s algorithm allow a quantum computer to factor the same number with lesser operations. However Shor’s algorithm has not gone so far in terms factoring large integers [1]. Shor’s algorithm is one of the few quantum that solves a computational problem with real-world applications: to efficiently find a factor q in a N, which is otherwise thought to be hard. In fact, the hardness of finding the integer factorization of a composite number, is one of the most widely believed conjectures in , and cryptographic applications that we all use in our daily life are built upon this [2]. Several experimental realizations of Shor’s algorithm for small numbers have been presented [3–9]. These are all very impressive demonstration of quantum optimal control, but experimental realization of Shor’s algorithm with the currently available technology is demanding, and this has led to the need for vast simplifications in the algorithm. Until now the largest number 143 factored on quantum computer device [10]. Only 4 qubits at 300K are used to perform this computation. Shor’s algorithm implemented so far using 4 qubits for factorization do not need prior knowledge of the answer [11-17]. Until 2012 the largest number factored using Shor’s algorithm was 15, and today the largest is still only 21. Since these are relied on Prior knowledge of the answer, the shor’s algorithm implementations were not genuine to the factorization problem. The idea of transforming factorization problem into optimization problem using quantum mechanics was introduced by Burges in 2001 [18], and further it was improved in 2010 by Schaller and Schutzhold [19].

2. FACTORIZATION PROBLEM

Finding prime factors of a given odd composite positive integer ‘n’. The problem of factoring integer ‘n’ can be reduced by choosing a random integer in relatively prime to n and finding smallest positive integer P such that

푚푃 = 1 푚표푑 푛 (1)

Shor developed the algorithm in polynomial time to solve the factorization problem. It consists of five steps. Among these steps the step 2 requires the use of quantum computer. Remaining steps of the algorithm can be performed in classical computer.

3. STEPS OF SHOR’S ALGORITHM

Step 1: Choose a random positive integer 푥 < 푛 . Compute the greatest common divisor gcd(x, n) using Polynomial time Euclidean algorithm. If 푔푐푑 (푥, 푛) ≠ 1, then proceed to next step.

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Step 2: Determine the unknown period P using quantum computer of the function 푓(푎) = 푥푎 푚표푑 푛 . Quantum computer is very efficient at period finding.

1 퐾 Step 3: If P is an odd integer, then go back to step 1. The probability of P being odd is ( ) , 2 where K is the number of distinct prime factors of n. If P is even, then proceed to step 4.

푃 푃 푃 Step 4: Since P is even, (푥 ⁄2 − 1) (푥 ⁄2 + 1) = 푥푃 − 1 = 0 푚표푑 푛. If (푥 ⁄2 + 1) = 푃 0 푚표푑 푛, then go to step 1. If (푥 ⁄2 + 1) ≠ 0 푚표푑 푛 then proceed to step 5. It can be shown 푃 1 퐾−1 that the probability that (푥 ⁄2 + 1) = 0 푚표푑 푛 is less than ( ) where k denotes the number 2 of distinct prime factors of n.

푃⁄ 푃⁄ Step 5: Using Euclidean algorithm compute 푑1 = 푔푐푑 (푥 2 − 1, 푛). Since (푥 2 + 1) ≠

0 푚표푑 푛, it can be easily shown that, 푑1 is a non trivial factor of 푛. Get another factor, 푑2 = 푛⁄ . 푑1

In the step 2 two quantum registers are initialized. The size of the register 1 is 퐿 qubits such that 푛2 ≤ 푞 = 2퐿 ≤ 2푛2. This register holds values {0, 1, 2, . . . . 푞 − 1 } and is used at the quantum fourier transform. Register 2 holds function values of 푓(푎) = 푥푎 푚표푑 푛. The size of the register 2 is 푀 = 푓푙표표푟(log2 푛) + 1.

|푅푒푔 1 > |푅푒푔 2 > = |푎 > |푓(푎) > = |푎 > |푏 >

= |푎0, 푎1, 푎2, . . . 푎퐿−1 > |푏0. 푏1, 푏2, . . . 푏푀−1 >

The step 2 of the shor’s algorithm is as follows.

Step 2.1: Initialize Register 1 and Register 2

|휓0 > =| 푅푒푔 1 > | 푅푒푔 2 > =|0 > |0 > = |0 0 0 0 . . . 0 > |0 0 0 0 . . . 0 >

Step 2.2: Apply the Q-point Fourier transform to register 1.

푞−1 푄퐹푇 1 |휓0 > → |휓1 > = ∑ |푎 > |0 > 푞 √ 푎=0

Register 1 holds all the integers 0, 1, 2, . . . 푞 − 1 in superposition.

Step 2.3: Let 푈푓 be the unitary transformation that |푎 > 푡표 |푎 > |푓(푎) > Apply the linear transformation 푈푓 to the two registers. The result is

푈푓 |휓1 > → |휓2 >

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푞−1 1 |푎 > |푓(푎) > |휓2 > = ∑ 푞 √ 푎=0

푞−1 1 |푎 > |푥푎 푚표푑 푛 > |휓2 > = ∑ 푞 √ 푎=0

The state of two registers is now more than a superposition of two states. In this step we have quantum entanglement of the two registers.

Step 2.4: Measure register 2. It will create a periodic superposition in register 1. Then |푓 > must collapse into some value 푓(푎0). The state of |푎 > will also collapse into the reimage of 푛 푓(푎 ). As 푓 is periodic, the reimage of 푓(푎 ) is {푎 , 푎 + 푃, 푎 + 2푃, . . . . + ( − 1) 푃}. 0 0 0 0 0 푟 We get,

푞−1 푚푒푎푠푢푟푒 푟푒푔푖푠푡푒푟 2 1 |푎 > |푓(푎) > → |휓3 > = ∑ 푞 √ 푎=0

푞−1 푝 |푎 + 𝑖푝 > |푓(푎 ) > |휓 > = √ ∑ 0 0 4 푞 푖=0

Step 2.5: Apply quantum Fourier transform to register 1.

푞−1 푝 푄퐹푇 |휓 > = √ ∑ |푎0 + 𝑖푝 > → 4 푞 푖=0

푞−1 푞 1 |𝑖 > Ø푖 |휓5 > = ∑ 푝 푝 √ 푖=0

Here, Ø푖 is some unimportant phase associated with each term due to linear shift 푎0.

푞 Step 2.6: Measure register 1. Measured value is a multiple of ⁄푝 . Save this value. 푞 Go back to step 2.1. Repeat the algorithm several times to get distinct multiples of ⁄푝. Once 푞 enough values are found, calculate the gcd to retrieve ⁄푝. As 푞 is known, 푝 is easily achieved.

4. FACTORING COMPOSITE NUMBER 풏 = ퟐퟏ

1) Choose 푥

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2) Determine 푞 3) Initialize first register (푟1) 4) Initialize second register (푟2) 5) Take QFT on first register 6) Measurement 7) Continued Fraction Expansion → Determine 푟 8) Check 푟 → Determine factors.

 If it is not co-prime with 푛, for example 푥 = 7, → 푔푐푑(푥, 푛) = 푔푐푑(7, 21) = 7 → 21 = 3 → 푑표푛푒, Factors are 7 and 3 7  If it is co-prime with 푛, for example 푥 = 11, → 푔푐푑(푥, 푛) = gcd(11, 21) = 1  Choose 푞 such that 푛2 ≤ 푞 ≤ 2푛2 → 푛2 = 212 = 441 , 2푛2 = 882, 푞 = 2퐿

Table 1. Selecting Appropriate 풒 value.

퐿 0 1 2 3 4 5 6 7 8 9 10

푞 1 2 4 8 16 32 64 128 256 512 1024

The number between 441 and 882 is 512. Hence 푞 = 512 = 29.

 Initialize registers 1 and 2. Thus the state of the two registers becomes

|휓0 > = |0 >푟1 |0 >푟2

511 1 |휓0 > = ∑ |푎 > |0 > 512 √ 푎=0

1 This corresponds to (|0 > +|1 >) on all bits. √2

 Initialize the 2nd register with superposition of all states 푥푎 푚표푑 푛.

511 1 푎 |휓1 > = ∑ |푎 > |11 푚표푑 21 > 512 √ 푎=0

Table 2. Finding Period P.

푎 0 1 2 3 4 5 6 7 8 9 . . .

11푎 푚표푑 푛 1 11 16 8 4 2 1 11 16 8 . . . Period 푃 = 6

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1 |휓1 > = { |0 > |1 > + |1 > |11 > + |2 > |16 > + |3 > |8 > + . . . . |511 > |11 >} √512

 Apply quantum Fourier transform on first register. The QFT is

푁−1 1 푘푛 푋(푘) = ∑ 푥(푛) 푊푁 √푁 푛=0

Put 푛 = 푚 푎푛푑 푁 = 512 in the above equation.

|휓2 > 푄퐹푇{|휓1 >}

511 1 푘푚 |휓2 > = ∑ |휓1 > 푊푁 512 √ 푚=0

511 511 1 1 푎 푘푚 |휓2 > = ∑ ∑ |푎 > |11 푚표푑 21 > 푊푁 512 512 √ 푚=0 √ 푎=0

Interchange the order of summation

511 511 1 |휓 > = ∑ |푎 > ∑ 푊 푎푚 |11푎 푚표푑 21 > 2 512 푁 푎=0 푚=0

For = 2,

511 1 |휓 > = |2 > ∑ 푊 2푚 |16 > 2 512 512 푚=0

Approximate result of above equation is 427. i.e C= 427.

 Continued Fraction Expansion:

퐶 427 1 = = 푎 + 푞 512 0 1 푎1 + 1 푎2 + 푎3+ . . . . .

푑0 = 푎0 푑1 = 1 + 푎0푎1 푑푛 = 푎푛푑푛−1 + 푑푛−2 푟0 = 1 푟1 = 푎1 푟푛 = 푎푛푟푛−1 + 푟푛−2

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427 1 1 1 1 1 = 0 + = 0 + = 0 + = 0 + = 0 + 512 512 85 1 1 512 1 + 1 + 1 + 427 푚표푑 512 427 427 427 2 5 + 85 85 1 1 = 0 + = 0 + 1 1 1 + 1 1 + 1 5 + 85 5 + 1 42 + 2 2

푑0 = 0 푑1 = 1 + 0 , 푟0 = 1, 푟1 = 푎1 = 1

푑2 = 푎2푑1 + 푑0 = 5 × 1 + 0 = 5

푟2 = 푎2푟1 + 푟0 = 5 × 1 + 1 = 6

푑3 = 푎3푑2 + 푑1 = 42 × 5 + 1 = 211

푟3 = 푎3푟2 + 푟1 = 42 × 6 + 1 = 253

푑4 = 푎4푑3 + 푑2 = 2 × 211 + 5 = 427

푟4 = 푎4푟3 + 푟2 = 2 × 253 + 6 = 512

Table 3. Finding 풓 through Continued Fraction Expansion.

푑 푑 211 푑 427 푛 0 1 2 3 4 1 = 1, 3 = , 4 = 푟1 푟3 253 푟4 512 푎 0 1 5 42 2 푛 푑 5 2 = → 푟 = 6 → 푇ℎ𝑖푠 푤표푟푘푠 푟 6 푑푛 0 1 5 211 427 2

푟푛 1 1 6 253 512

푃⁄ 6⁄ First Factor of 21 is 푑1 = 푔푐푑 (푥 2 − 1, 푛) = 푔푐푑 (11 2 − 1, 21) = 푔푐푑(1330, 21) = 7

푛 21 Second factor of 21 is 푑2 = = = 3 푑1 7

5. FACTORING COMPOSITE NUMBER OF 풏 = ퟗퟏ

Choose 푞 such that 푛2 ≤ 푞 ≤ 2푛2. 8281 ≤ 푞 ≤ 16562, 퐿푒푡 푞 = 214 = 16384 Step 1: Choosing random positive integer 푥 = 3, Since 푔푐푑(91, 3) = 1, proceed to step 2 to find the period 푃.

푓(푎) = 푥푎 푚표푑 푛 = 3푎 푚표푑 91

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Table 4. Selecting Period P.

푎 0 1 2 3 4 5 6 7 . . . . . 푓(푎) 1 3 9 27 81 61 1 3 . . . . . Period P = 6

Step 2: Initialize registers 1 and 2. Thus the state of the two registers becomes

|휓0 > = |0 > |0 >

16383 1 |휓0 > = ∑ |푎 > |0 > √16384 푎=0 1 This corresponds to (|0 > +|1 >) on all bits. Initialize the 2nd register with √2 superposition of all states 푥푎 (푚표푑 푛). 16383 1 푎 |휓1 > = ∑ |푎 > |3 (푚표푑 푛) > √16384 푎=0

16383 1 푎 |휓1 > = ∑ |푎 > |3 (푚표푑 91) > √16384 푎=0 1 |휓1 > = {|0 > |1 > + |1 > |3 > +|2 > |9 > +|3 > |27 > + |4 > |81 > +|5 √16384 > |61 > +|6 > |1 > +|7 > |3 > +|8 > |9 > +|9 > |27 > +|10 > |81 > +|11 > |61 > + |12 > |1 > +|13 > |3 > + ...... + |16380 > |1 > +|16381 > |3 > + . . . . |16383 > |27 >}

Step 3: The quantum Fourier transform on the 1st register The QFT is 푁−1 1 푘푚 푋(푘) = ∑ 푥(푛)푊푁 √푁 푚=0

16383 16383 1 1 푎푚 푎 |휓2 > = ∑ ∑ 푊푁 |푎 > |3 (푚표푑 91) > √16384 √16384 푚=0 푎=0 Consider,

16383 푎푚 푎 푚 2푚 3푚 ∑ 푊푁 |푎 > |3 (푚표푑 91) > = |1 > +푊푁 |3 > +푊푁 |9 > +푊푁 |27 푎=0 4푚 5푚 16380푚 16381푚 > +푊푁 |81 > +푊푁 |61 > + . . . . +푊푁 |1 > +푊푁 |3 16383푚 > + . . . . 푊푁 |27 >

The result of |휓2 > = 퐶 = 13453.  Continued Fraction Expansion:

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퐶 13453 1 1 1 1 = = 0 + = 0 + = 0 + = 0 + 16384 2931 1 1 푞 16384 1 + 1 + 1 + 13453 13453 13453 1729 4 + 2931 2931 1 1 1 = 0 + = 0 + = 0 + 1 1 1 1 + 1 1 + 1 1 + 1 4 + 2931 4 + 1202 4 + 1 1729 1 + 1729 1 + 1729 1202 1 1 = 0 + = 0 + 1 1 1 + 1 1 + 1 4 + 1 4 + 1 1 + 527 1 + 1 1 + 1 + 1202 1202 527 1 1 = 0 + = 0 + 1 1 1 + 1 1 + 1 4 + 1 4 + 1 1 + 1 1 + 1 1 + 148 1 + 1 2 + 2 + 527 527 148 1 = 0 + 1 1 + 1 4 + 1 1 + 1 1 + 1 2 + 83 3 + 148 1 = 0 + 1 1 + 1 4 + 1 1 + 1 1 + 1 2 + 1 3 + 1 1 + 1 1 + 1 3 + 1 1 + 1 1 + 1 1 + 1 1 + 3

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Table 5. Finding 푟 through Continued Fraction Expansion.

푛 0 1 2 3 4 5 6 7 8 9 10 11 12 13

푎푛 0 1 4 1 1 2 3 1 1 3 1 1 1 1

푑푛 0 1 4 5 9 23 78 101 179 638 817 1455 2272 3727

푟푛 1 1 5 6 7 20 67 87 154 549 703 1252 1955 3207

푑0 = 푎0 푑1 = 1 + 푎0푎1 푑푛 = 푎푛푑푛−1 + 푑푛−2

푟0 = 1 푟1 = 푎1 푟푛 = 푎푛푟푛−1 + 푟푛−2

푃⁄ 6⁄ First Factor of 91 is 푑1 = 푔푐푑 (푥 2 − 1, 푛) = 푔푐푑 (3 2 − 1, 91) = 푔푐푑(26, 91) = 13

푛 91 Second factor of 21 is 푑2 = = = 7 푑1 13

6. EUCLIDEAN ALGORITHM

Euclidean algorithm is an efficient method for computing the greatest common (gcd). Algorithm steps are as follows:

Step 1: Let 푎 and 푏 are the integers such that 푎 > 푏 and 푟 be the remainder when 푎 is divided by 푏. Assume 푟 ≠ 0 and 푞 is the quotient.

Step 2: 푎 = 푏푞 + 푟, 푟 = 푎 − 푏푞 . Whatever divides 푎 and 푏 must also divide 푟 and also divide 푏. Similarly any divisor of 푏 and 푟 must also be divisor of 푎 and also divide 푏. Thus 푔푐푑(푎, 푏) = 푔푐푑(푏, 푟). When 푟 = 0, 푎 = 푏푞 then 푔푐푑(푎, 푏) = 푏. Algorithm works by repeatedly using 푔푐푑(푎, 푏) = 푔푐푑(푏, 푟). If 푎 > 푏 then compute remainder 푟 표푓 푎 ÷ 푏. If this is zero, then 푔푐푑(푎, = 푏) = 푏. If 푟 ≠ 0, take 푏 and 푟, compute ′ the remainder 푟1 = 푏 ÷ 푟. If 푟1 = 0 then using 푔푐푑(푎, 푏) = 푔푐푑(푏, 푟) = 푔푐푑(푟, 푟 ). Continue until the remainder is zero.

7. CONCLUSIONS

Mathematical description on factorization of composite numbers 푛 = 21 and 푛 = 91 are cleared explained with all steps using shor’s algorithm. Quantum part in the factorization problem is shown with approxiamte results. This helps in finding the value of 푟. Using this value factors compoiste numbers are determined with the help Euclidean algorithm. The mathematical description explained in this paper can be used to factorize the larege integers.

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References

[1] Nikesk S Dattani and Nathaniel Bryans, Quantum factorization of 56153 with only 4 qubits, Quantum Physics, 27th November 2014. arXiv:1411.6758 [2] Niklas Johansson and Jan Ake Larsson, Realization of Shor’s Algorithm at Room Temperature, Quantum Physics, 10th January 2017. arXiv:1706.03215 [3] T. Monz et al., Realization of a scalable Shor algorithm, Science 351, 1068–1070 (2016). [4] E. Lucero et al., Computing prime factors with a Josephson phase qubit quantum processor. Nature Physics 8, 719-723 (2012). [5] E. Martin-Lopez et al., Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature Photonics 6, 773–776 (2012). [6] Politi, J. C. Matthews, and J. L. O’Brien, Shor’s quantum factoring algorithm on a photonic chip. Science 325, 1221–1221 (2009). [7] C.-Y. Lu, D. E. Browne, T. Yang, and J.-W. Pan, Demonstration of a Compiled Version of Shor’s Quantum Factoring Algorithm Using Photonic Qubits. Physical Review Letters 99 (2007) 10.1103/PhysRevLett.99.250504 [8] B. P. Lanyon et al., Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement. Physical Review Letters 99 (2007) 10.1103/PhysRevLett.99.250505 [9] L. M. Vandersypen et al., “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001). [10] N. Xu, J. Zhu, D. Lu, X. Zhou, X. Peng, and J. Du. Quantum Factorization of 143 on a Dipolar-Coupling Nuclear Magnetic Resonance System. Physical Review Letters, vol. 108, no. 13, p. 130501, Mar. 2012. [11] L. M. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature, vol. 414, no. 6866, pp. 883–7, Jan. 2001. [12] B. Lanyon, T. Weinhold, N. Langford, M. Barbieri, D. James, A. Gilchrist, and A. White, Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement. Physical Review Letters, vol. 99, no. 25, p. 250505, Dec. 2007. [13] C.-Y. Lu, D. Browne, T. Yang, and J.-W. Pan, Demonstration of a Compiled Version of Shor’s Quantum Factoring Algorithm Using Photonic Qubits. Physical Review Letters, vol. 99, no. 25, p. 250504, Dec. 2007. [14] Politi, J. C. F. Matthews, and J. L. O’Brien, Shor’s quantum factoring algorithm on a photonic chip. Science, vol. 325, no. 5945, p.1221, Sep. 2009. [15] E. Martín-López, A. Laing, T. Lawson, R. Alvarez, X.-Q. Zhou, and J. L. O’Brien, Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature Photonics, vol. 6, no. 11, pp. 773–776, Oct. 2012.

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Biography

Poornima Aradhyamath received B.Sc degree from Gulbarga University and M.Sc in Physics from Karnataka University Dharawad, Karnataka. She worked as assistant professor at BITM, for 10 years. Presently she is a regular research scholar in the Department of Physics at RYM Engineering College, Ballari. She has four research paper in international Journals and presented two papers in international conferences.

Nagabhushana N. M. received B.Sc, M.Sc and PhD degree from Gulbarga University, Karnataka. His area of research is Nuclear Physics, Material and . He has Published 10 research papers in International journals and many papers in national and international conferences. Presently he is working as a professor and HOD of physics department at RYM Engineering College, Ballari, Karnataka. He has guided many M.Sc projects and guiding 8PhD students under VTU, Belagavi

Rohitha Ujjinimatad received B.E. degree in Electronics and Communication Engineering from Bangalore University, M.Tech. in Digital Electronics and Ph.D. from VTU, Belagavi, India. He is student member of IEEE and life member of ISTE and he is working as Professor in ECE Department at Proudadevaraya Institute of Technology, Hospete, India. His area of research is spectrum sharing and spectrum sensing in cognitive radio networks. He has guided many B.E. and M.Tech. Projects, and guiding five Ph.D. students. He has one national and 16 International Conference Papers to his credit. Also he has published papers in IET Communications, IET Journal of Engineering, Springer, IJCAand other reputed journals etc.

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