1. Introduction

2. A Novel Security System on the Internet of Things

3. Modified Elliptic Curve Cryptography Algorithm

• Step 4. Calculate the public key,

• Step 5. Calculate K value:

• Step 6. Apply the encryption equation.

• Step 7. Send that ciphertext to the destination.
• Step 8. Receive that ciphertext.
• Step 9. Apply the decryption equation.
  
• Step 10. Mapping the message back.

Full Educational Example

In this example we are going to encrypt message M={B} using modified elliptic curve algorithm with an improved PUF device. Use the following element values:

$y^2=x^3+2x+2 \left(Mod 17\right)$P=17

a = 2 & b = 2

$G\left(x_1, y_{1 }\right)=\left(5, 1\right)$n = 19

h = 1

The solution:

Step 1. Encoding the Plaintext & Mapping the message on the curve

Let’s assume {B} character equal { 11 } in ASCII table

Choose integer k as auxiliary base parameter {20}

Compute x = m * k + 1 = 11 * 20 + 1 = 221

Compute $y^2=x^3+ax+b \left(Mod P\right)$

y2 = (221)3 + [2*221] + 2 Mod 17

y2= 4

The message mapped on the curve on point (221, 4)

Step 2. Generating the Points

G (5, 1) → $1G\left(X_{1G} , Y_{1G}\right)$

2G = G + G we have to apply a point doubling method

$\lambda ={\displaystyle \frac{3x_{1G}^2+a}{2y_{1G}}}={\displaystyle \frac{3* 5^2+2}{2*1}}={\displaystyle \frac{77}{2}}=77* 2^{-1}=81 Mod 17=13$

$2^{-1}\to 2*\mathit{X}=1 mod 17$

X is a number between {1; n-1} And when multiply it with 2 mod 17 will equal 1

Solution: X=9 because (2 * 9) mod 17 = 1 → $\lambda =\left(9*9\right)mod 17=13$

$x_{2G}={\lambda }^2-2x_{1G}={13}^2-\left(2*5\right)=16-10=6 Mod 17=6$

$y_{2G}=\lambda \left( x_{1G}-x_{2G} \right)-y_{1G}=13\left(5-6\right)-1=-14 Mod 17=3$

2G= (6,3)

Now we have 1G (5,1) & 2G(6,3) → Let’s generate 3G Note. 1G != 2G

3G = 1G + 2G → $3P=\left(x_{2G}, y_{2G}\right)+\left(x_{3G}, y_{3G}\right)$ Where $3G=\left(x_{3G}, y_{3G}\right)$

if 1G != 2G → $\lambda ={\displaystyle \frac{y_{2G}-y_{1G}}{x_{2G}-x_{1G}}}={\displaystyle \frac{3-1}{6-5}}={\displaystyle \frac{2}{1}}=2$

$x_{3G}={\lambda }^2-x_{1G}-x_{2G}=2^2-5-6=-7 mod 17=10$ $y_{3G}=\lambda \left( x_{1G}-x_{3G} \right)-y_{1G}=2\left(5-10\right)-1=-11 mod 17=6$

3G= (10,6) And thus we can calculate all the points,

Total Points = n - 1 = 19 - 1 = 18 points

Step 3. Calculate private key.

Sending a challenge to PUF device to generate an advanced random number. Once the random number is ready, the algorithm can calculate the private number.

$\mathrm{d}=R^3+a^2+b Mod p$

R: Random number comes from PUF device

$\mathbf{a}, \mathbf{b}$ Two elements on the curve.

p: Specifying the finite field F_p

Next step is Generate the public key:

Step 4. Calculating the public key:

Random number (d) in the interval [1; n - 1].

Compute Q = d G. where.

Q: Public Key

d: Private key

G: Generator Point

Let’s specify a private key d For Example d=5.

Q=5G → (9, 16) It is a public key.

Step 5. Calculate K value:

Sending a challenge to PUF device to generate an advanced random number. Once the random number is ready, the algorithm can calculate the K value.

K = $R^3+a^2+b Mod p$

Step 6. Applying the encryption equation:

Compute

G: Generator Point

M: Plaintext.

Q: Public key

$Ciphertext \left(C_1, C_{2 }\right)$

$C_1=\left[k\right]P$

$C_2=M+\left[k\right]Q$

-Let’s assume k ∈ [1; n-1] Let’s say k= 13

P is the base point (5, 1).

Q is the public key (9, 16).

M is the message mapped on the curve (221, 4).

$C_1=13\left(5, 1\right)=\left(16, 14\right)$

$C_2=\left(221, 4\right)+\left[13\right]\left(9,16\right)=\left(7, 8\right)$

$Ciphertext \left(\left(16, 14\right), \left(7, 8\right)\right)$

Step 7 & 8 will be transmitting & receiving the ciphertext to the destination.

Step 9. Applying the decryption equation:

$\mathbf{M}=\left(7, 8\right)-\left[5\right]\left(16. 14\right)$

$M=\left(221, 4\right)$

Step 10. Mapping the message back

It is the last step, map the point back by computing (x - 1) / K

K=20, x =221

(221 – 1) / 20 = 11

Decoding the plaintext by ASCII code 11 = ‘B’ for example.

4. The Authority Device (AD)

5. The Node Device

6. The System Reliability and Integrity

7. Implementing the Modified ECC Algorithm on Internet of Things

8. A Comparison Study between the Standard and the Novel Security System

9. Conclusions

References

1. Controlled Physical Random Function .Blaise Gassend, Dwaine Clarke, Marten van Dijk † and Srinivas Devadas. 13 Dec. 2002, 18th Annual Computer Security Applications Conference, 2002. Proceedings., pp. 149-160.
2. Goutsos, Konstantinos.Physical Unclonability Framework forthe Internet of Things. The UK - Newcastle : Newcastle University, 2019.
3. KINZA SHAFIQUE, BILAL A. KHAWAJA, FARAH SABIR, SAMEER QAZI, MUHAMMAD MUSTAQIM. Internet of Things (IoT) for Next-Generation Smart Systems: A Review of Current Challenges, Future Trends and Prospects for Emerging 5G-IoT Scenarios. s.l. : IEEE Acess, 2020.
4. Internet of Things security: A survey.Fadele Ayotunde Alabaa, Mazliza Othman, Ibrahim Abaker Targio Hashem, Faiz Alotaibi. 2017, Journal of Network and Computer Applications, Vol. 88, pp. 10-28.
5. András Varga, Rudolf Hornig.AN OVERVIEW OF THE OMNeT++ SIMULATION ENVIRONMENT. Budapest, Hungary : s.n., 2016.
6. (Ed.), Phillip Rogaway.Advances in Cryptology – Crypto 2011. Santa Barbara, CA, USA : Springer, August 14-18, 2011.
7. Niels Ferguson, Bruce Schneier, Tadayoshi Kohno.Cryptography Engineering: Design Principles and Practical Applications. s.l. : Wiley 1st edition, March 2010.
8. Elliptical Curve Cryptography Design Principles.J VenkataGiri, Dr. ASR Murty. 2021. International Conference on Recent Trends on Electronics, Information, Communication & Technology (RTEICT).
9. Analysis of Encryption Algorithms Proposed for Data Security in 4G and 5G Generations.Khalid Fadhil Jasim 1*, Kayhan Zrar Ghafoor2,3, and Halgurd S. Maghdid. 2022. ITM Web of Conferences 42, 01004 (2022).
10. Analysis of Standard Elliptic Curves for the Implementation of Elliptic Curve Cryptography in Resource-Constrained E-commerce Applications. Javed R. Shaikh, Maria Nenova, Georgi Iliev and Zlatka Valkova-Jarvis. Sofia, Bulgaria : s.n., 2017. IEEE International Conference on Microwaves, Antennas, Communications and Electronic Systems (COMCAS).
11. Elliptic Curve Cryptosystems.Koblitz, Neal. 1987, MATHEMATICS OF COMPUTATION, Vol. 48, pp. 203-209.
12. Constandinos X. Mavromoustakis, Jordi Mongay Batalla, George Mastorakis.Internet of Things (IoT) in 5G Mobile Technologies. Switzerland : Springer International Publishing Switzerland , 2016.
13. Lightweight and Secure PUF Key Storage Using Limits of Machine Learning.Meng-Day, Yu1, David M’Raihi, Richard Sowell,. 2011, International Association for Cryptologic Research, p. 16.
14. Brown, Daniel R. L.SEC 1: Elliptic Curve Cryptography. s.l. : Standards for Efficient Cryptography, 2009.
15. SEC 2 : Recommended Elliptic Curve Domain Parameters. s.l. : Standards for Efficient Cryptography, 2010.
16. Buckley, Eoin. SEC 4 Elliptic Curve Qu-Vanstone Implicit Certificate Scheme (ECQV) - draft 2. s.l. : Standards for Efficient Cryptography, 2014.
17. Silicon Physical Random Functions.Blaise Gassend, Dwaine Clarke, Marten vandijk, and Srinivas Devadas. November 2002. Proceedings of the 9th ACM conference on Computer and communications security.
18. Implementation of ElGamal Elliptic Curve Cryptography Over Prime Field Using C.Debabrat Boruah, Monjul Saikia. India : s.n., 2014. ICICES2014 - S.A.Engineering College, Chennai, Tamil Nadu, India.
19. The State of Elliptic Curve Cryptography.NEAL KOBLITZ, ALFRED MENEZES, SCOTT VANSTONE. 2000, Kluwer Academic Publishers, Boston. Manufactured in The Netherlands, Vol. 19, pp. 173–193.
20. Towards 5G Security Analysis against Null Security Algorithms Used in Normal Communication.Run Zhang,1 WenAn Zhou ,1 and Huamiao Hu2. 2021, Hindawi Security and Communication Networks, p. 15.
21. Buckley, Eoin.SEC 4: Elliptic Curve Qu-Vanstone Implicit Certificate draft - 1. s.l. : Standards for Efficient Cryptography, 2013.
22. AES, DES, and RSA: A Comprehensive Study on Data Security Mechanisms.Kapoor, Aditya. 2018, IJCAM Research Consultant, Indore , India, p. 9.
23. Maletsky, Kerry.RSA vs ECC Comparasion for Embedded system. s.l. : Microchip, 2020.
24. Shaheen, Mai Helmy. Hybrid Encryption Algorithms Over Wireless Communication Channels. Oxon : A SCIENCE PUBLISHERS BOOK, 2022.
25. Advances in Security Technology, Security Analysis of “A Novel Elliptic Curve Dynamic Access Control System”. Haeng-kon Kim, Tai-hoon Kim, Akingbehin Kiumi. China : s.n., 2008. International Conference SecTech.
26. Cryptographic Analysis of DES and RSA Algorithm Using the AVISPA Tool andWSN.Shailendra Singh Gaur, Megha Gupta, and Gautam Gupta. 2021, Proceedings of 3rd International Conference on Computing Informatics and Networks, Vol. 167.
27. Tom St Denis, Simon Johnson.Cryptography for Developers. s.l. : Syngress, 2006.
28. Sumit Singh Dhanda, Brahmjit Singh, and Poonam Jindal.Elliptic Curve Cryptography: A Software Implementation. India : Department of Electronics and Communication Engineering, National Institute of Technology, 2021.
29. Encoding And Decoding of a Message in the Implementation of Elliptic Curve Cryptography.Padma Bh, D.Chandravathi , P.Prapoorna Roja. 2010, (IJCSE) International Journal on Computer Science and Engineering, Vol. 02, p. 05.
30. Burt Kaliski, Terry S. Arnold,.IEEE Standard Specifications for Public-Key Cryptography. New York : American National Standards Institute, 2000.