review on the memory efficient rsa variants - International Journal of ...

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology

ISSN: 0975-766X CODEN: IJPTFI Review Article

Available Online through www.ijptonline.com REVIEW ON THE MEMORY EFFICIENT RSA VARIANTS

Chandra Segar Thirumalai* School of Information Technology and Engineering, VIT University, Vellore – 632014. Email: [email protected] Received on 12-09-2016 Accepted on: 02-11-2016 Abstract Public key cryptography (PKC) plays the phenomenal role in asymmetric based security applications such as E-mail, File sharing, Cloud storage, etc. As like in std. RSAet. al [11], Dual RSA scheme et. al [6] has public code  E, N1, N2  and private code  d , r1, r2 , s1, s2  . In this paper Trivial RSA is realized with  E, N1 , N2 , N3  and  D, r1 , r2 , r3 , s1 , s2 , s3  codes which essentially satisfies the condition ED  1  z1 N1  1  z2 N 2  1  z3 N 3 . Here a1 , a2 , b1 , b2 interrelated keys are used to generate primes r1 , r2 , r3, s1 , s2 , s3 which takes z2  b1 and z3  b2 . The strength of keys are tested using lattice constraint nd  n 0.33 . Keywords: Trivial PKC, Lattice, RSA, PKC, Encryption, Decryption I. Introduction Cryptosystem is one of the oldest and finest techniques for data security and information hiding. Now days RSA PKC is widely used for many security based applications. But, by using quantum cryptography, the keys of Std. RSA might get broken down around 850 bits. These result in the need for the enhancement in current PKC cryptosystem. Some of the variants of RSAet. al [1,2,5,6,8,12,13,14,19,22,28]are proposed to meet this demand. Dual RSA is one such variant of RSA in which two keys can be generated with sharing N moduli and this saves the memory requirements of the key storage. Here, in our Trival RSA, three keys are generated at same time in KGS (Key Generation System) phase, which primarily aims to reduce the generation time and also to strengthen the keys by using lattice method. The strength of these keys can be analysed by using lattice constraint nd  n 0.33 . From which only the strong keys will be shared among users there by makes the system unbreakable.

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4907

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology In RSA based PKC, the KGS generates the public and private key is shown in the following figure 1. in which the public key is shared among the user for to encrypt their corresponding confidential message

into cipher text

and

this process is called encryption (C  M e mod N ) . Then the corresponding user will revert back this encrypted message (cipher) to server or KGS. Using the private key, the KGS will retrieves back the original message cipher

and this process is called decryption (M  C d mod N ) .Here the

random primes

from the received

-bit moduli is trivially derived on two balanced

Using the Euler totient function  (n) calculated by ( p  1)*(q 1) , the public key

is selected wisely

with the constraint such that 1  e   (n) and gcd (e, (n))   1 . Finally in KGS phase, the private key is generated using Extended Euclid algorithm using these  (n), e which is shown in the following Table 1.Suppose the KGS phase selects the

,

and  (n) =89964.

then

Table-1: Findind ‘D’ through extended euclid’s algorithm

Here in Table 1.

q

r1=  (n)

r2=e

r

t1

t2

t=t1-qt2

4

89964

17993

17992

0

1

-4

1

17993

17992

1

1

-4

5

17992

17992

1

0

-4

5

-89964

0

1

0

-

5

-89964

5

and are quotient, remainder and temporary variables respectively. In the last row and last columns

gives us the private key

which is the multiplicative inverse of the public key Sender side:

Original Message to be sent to Receiver side

1-a Receives Public key n,e

1-b Sender’s Public key

1-c To Encrypt the Message with Public Key

2 Encrypted Messages

Receiver side:

1. Key Generator-n,e,d

3-a Receiver’s private Key d

3-bTo Decrypt the Encrypted Messages with Private Key algorithm 3-c Decrypted Messages 3-d Original Messages

Figure. 1 Architecture of RSA Public Key Cryptosystem.

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4908

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology II. Literature Review K. Suresh et. al [1] investigates on Batch RSA, N-prime RSA and rebalanced RSA with respect to its key generation, encryption and decryption respectively. Here they recommends the security safety measure based on their variants operation such as for rebalanced RSA the private key exponent attack possibility is less as its private key size is quite larger than its variants. The KGS phase computation of Batch RSA and std. RSA are similar in generating the keys. In case of N-prime RSA, as the bit length of N-bit moduli is common with its variants, as its N-prime factors increases then its bit length gets decreases, finally the security strength might get vulnerable toward the factorization attack. Whereas the rebalanced RSA key generation takes 1.27 times more than the std. RSA since it takes additional two random primes. In case of encryption, the Batch RSA time is efficiently improving along the group of public keys whereas, in Rebalanced RSA time gets increases since its public key size is comparatively larger than its variants. In case of decryption, the Rebalanced RSA is more efficient than its variants, since its private key d=(n/2)*(1/k) whereas the Nprime RSA d=(n/2), in Batch RSA and Std. RSA d=n bits. However, the Batch RSA runs in parallel with the system, hence it takes lesser time than the std. RSA decryption. Mayank Jhalaniet. al [4] modified the N-prime RSA and claims it as multi power RSA for to improve the decryption cost by using common private key d. Here the N-bit moduli is calculated as N=pk-1*q. Then the Euler totient function and then e and d are selected in the usual way. Thefinal private keys (dp, dq, p, q) are generated in such a way that, dp=d mod (p-1) and dq=d mod (q-1). Here the cipher text C is generated by breaking the message M into two pieces. Now, for decryption the following ways are adopted: M1=C1dp mod pk-1 M2=C2dq mod q Partial Key Exposure Attack on RSA Using a 2-Dimensional Latticeet. al [37] are illustrated below: Algorithm: 1. Get the values of N, e. ~

2. We know some MSB of d, take that approximation as d . ~

3. Then d0  d  d .

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4909

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology [ n (    0.5)]

4. Calculate C  2 5. Calculate R=e.d

6. Compute { d 0 , k }from

d   C0   d0   0  r.  0   v    .     e  N  k    k    R    7. Calculate   N  

ed  1 k

8. Calculate pand q from   N  . III. Proposed system In this paper a new variant of RSA is realized which can generate three public keys at the same time and so it is Trivial RSA. In this method four random numbers a1 , a2 , b1 , b2 are used to generate r1 , r2 , r3, s1 , s2 , s3 primes which are interrelated to each other. Public  E, N1 , N2 , N3  and private  D, r1 , r2 , r3 , s1 , s2 , s3  codes which essentially satisfies the condition

ED  1  z1 N1  1  z2 N 2  1  z3 N 3 where z2  b1 and z3  b2 . Then the strength of keys are tested using lattice constraint nd  n 0.33 .

Figure. 2 Architecture of Trivial RSA Public Key Cryptosystem. Algorithm: 

KGS Phase: 1. Randomly select a1 , a2 such that r1  a1a2  1 where r1 is prime. 2. Randomly select b 2 such that r2  a1b2  1 where r2 is prime. 3. Randomly select b1 such that r3  a2b1  1 where r3 is prime.

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4910

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology z2  b1 , s3  z1a1  1 and z3  b2 4. Compute s1  b1b2  1 , s2  z1a2  1 , such a way that

ED  1  z1 N1  1  z2 N 2  1  z3 N 3 where s1 , s2 , s3 are primes, and N1  a1b1 , …, N 3  a3b3 . 5. And compute D from above equation. Output: Public key  E, N1 , N2 , N3  and Private key  D, r1 , r2 , r3, s1 , s2 , s3  . 

Encryption: Input

: Message M i and public key E.

Process: Compute, Ci  M i mod N i where i=1,2,3. E

Output: Cipher text, C i . 

Decryption: Input: C i , private key: ( D, N1 , N 2 , N 3 ) Process: Compute, Pi  CiD mod N i Output: Plain text, Pi .

Trivial RSA Instance: KGS: a1  20 a2  9 b1  8 b2  12 r1  a1a2  1  181 r2  a1b2  1  241 r3  b2 a1  1  73 s1  b1b2  1  97 s2  z1a2  1  19 s3  z1a1  1  41 z 2  b1  8 z3  b2  12 z1  2

E  17 D  2033 ED  1  z1 N1  1  z2 N 2  1  z3 N3  34561 Public key  E, N1 , N2 , N3  =(17,17557,4579,2993)

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4911

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology Private key  D, r1 , r2 , r3, s1 , s2 , s3  =(2033,181,241,73,97,19,41) Encryption: Cipher text: Ci  M i mod N i where i=1,2,3. E

M 1  267, M 2  167, M 3  356 C1  26717 mod 17557

= 15996 C2  16717 mod  4579

= 451 C3  35617 mod  2993

= 2636 Decryption: Plain text: Pi  CiD mod N i

P1  159962033 mod 17557 = 267 P2  4512033 mod  4579

= 167

P3  26362033 mod  2993 = 356 IV. Performance Analysis Table-2: Comparison of Encryption and Decryption of RSA, Dual RSA and Trivial RSA. N-Bit

Encryption RSA

Decryption

Dual

Trivial

RSA

RSA

RSA

Dual

Trivial

RSA

RSA

512

6

7

10

7

13

20

1024

29

47

70

47

93

139

2048

178

352

547

361

702

1085

4096

1456

2963

4041

2664

5250

8063

8192

10419

23338

27108

13211

32406

61593

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4912

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology Table-3: Overall Performance Comparison of Encryption and Decryption time of RSA, Dual RSA and Trivial RSA. Encryption Performance Comparison

Decryption Performance Comparison

Dual RSA

Trivial RSA

Dual RSA

Trivial RSA

vs

vs

vs

vs

RSA vs

RSA vs

Dua

Dua

Dua

l

l

Trivi

Trivi

l

RS

Trivial

RS

Trivia

RS

Dual

RS

al

RS

al

RS

RS

A

RSA

A

l RSA

A

RSA

A

RSA

A

RSA

A

A

0.45

0.38

2.2

0.84

2.62

1.18

0.42

0.22

2.36

0.54

4.35

1.84

The performance of both encryption and decryption are shown in the above Table 2. Also its overall performance is given in Table 3 for comparative study. From the result, it has observed that trivial RSA and Dual RSA encryption goes together whereas in decryption trivial RSA takes around double the time of Dual RSA. V. Conclusion As the N-bit moduli of PKC increases, the variants also reflect towards their transformation time. This is mainly happens due to the production of its related keys with sharing the safe private key size nd  n 0.33 . The main advantage of this dual and trivial RSA is to optimize the memory storage size, since its public and private keys are commonly shared among the n-users. Moreover, this multi user secure scheme of sharing the common public and private key with small N-bit moduli can be used to authenticate the user over a short span of time say OTP (One time password) acknowledgement of Banking application, session time based application and so on effectively to get rid of quantum computing based attack. In future, using parallel computing and efficient library methods we can attain efficient result at higher N-bit moduli to increase the users. References: 1.

K. Suresh, K.Venkataramana, “Study of analysis on RSA and its variants” in International Journal of Computer Science Research & Technology, vol. 1, issue 4, 2013

2.

RohitMinni, Kaushal Sultania, Saurabh Mishra, Durai Raj Vincent, “Enhancing security features in RSA cryptosystem” in Computing, Communications and Networking Technologies, 2013.

3.

C.C. Chang and M.S. Hwang “Parallel computation of the generating keys for RSA cyptosystems”. IEEE 1996

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4913

4.

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology Mayank Jhalani, Piyush Singh, Gaurav Shrivastava,“ Enhancement over the variant of public key cryptography algorithm,” in International journal of emerging technology and advanced engineering, Vol. 2, Issue 12, Dec. 2012

5.

NavaneetOjha, SahadeoPadhye “Cryptanalysis of multi prime RSA with secret key greater than public key,” International Journal of network security, vol.16, no.1, pp.53-57, Jan. 2014

6.

Hung-min sun, Mu-en wu, Wei-chi ting, and M. Jason Hinek “Dual RSA and its security analysis,” IEEE transactions on information theory, vol. 53, no. 8, august 2007.

7.

B R Ambedkar & S S Bedi, “A New Factorization Method to Factorize RSA Public Key

Encryption”, IJCSI

International Journal of Computer Science Issues, Vol. 8, November 2011. 8.

Chhabra A, Mathur S., 2011, “Modified RSA algorithm: a secure approach”. In: International Conference on Computational Intelligence and Communication Networks, Gwalior; 2011.

9.

Forouzan BA.2007, “Cryptography and network security”. Special Indian Edition. Tata McGraw-Hill, p. 2011.

10. Ali H, Salami MA. 2004, “Timing attack prospect for RSA cryptanalysts using genetic algorithm technique”. Int Arab J Inf Technol 2004. 11. Rivest RL, Shamir A, Adleman LA.1978, “Method for obtaining digital signatures and public-key cryptosystems”. Commun ACM. 12. Hung-min sun, mu-en wu, wei-chi ting, and m. jason hinek “Dual RSA and its security analysis”. IEEE transactions on information theory, vol. 53, no. 8, august 2007. 13. Ximeng Liu, Jianfeng Ma, Jinbo Xiong, Qi Li, Tao Zhang, Hui Zhul “Threshold attribute-based encryption with attribute hierarchy for lattices in the standard model”-IET 2013 14. Thangavel, M., P. Varalakshmi, MukundMurrali, K. Nithya,“An Enhanced and Secured RSA Key GenerationScheme (ESRKGS),” in Journal of information Security and application, Vol. 20, 2015, pp. 3-10. 15. Ravi Shankar Dhakar, Amit Kumar Gupta, Prashant Sharma, 2012,” Modified RSA Encryption Algorithm (MREA)” advance Advanced Computing & Communication Technologies (ACCT). 16. Chandrasegar Thirumalai, Senthilkumar M, “An Assessment Framework of Intuitionistic Fuzzy Network for C2B Decision Making”, International Conference on Electronics and Communication Systems (ICECS), 2016. 17. “DDoS: Survey OfTraceback Methods”, International Joint Journal Conference in Engineering 2009, ISSN 17979617. IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4914

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology 18. Vaishnavi B, Karthikeyan J, Kiran Yarrakula, Chandrasegar Thirumalai, “An Assessment Framework for Precipitation Decision Making Using AHP”, International Conference on Electronics and Communication Systems (ICECS), 2016 19. M.Senthilkumar,T.Chandrasegar, M.K. Nallakaruppan, S.Prasanna,“A Modified and Efficient Genetic Algorithm to Address a Travelling Salesman Problem,” in International Journal of Applied Engineering Research,Vol. 9 No. 10, 2014, pp. 1279-1288 20. Vinothini S, Chandra Segar Thirumalai,,Vijayaragavan R, Senthil Kumar M,“A Cubic based Set Associative Cache encoded mapping International Research Journal of Engineering and Technology (IRJET),” Volume: 02 Issue: 02 May -2015 21. T Chandra Segar, R Vijayaragavan,“Pell's RSA key generation and its security analysis,”in Computing, Communications and Networking Technologies(ICCCNT)2013, pp. 1-5. 22. Chandramowliswaran N, Srinivasan.S and Chandra Segar.T, “A Novel scheme for Secured Associative Mapping” The International J. of Computer Science and Applications (TIJCSA) & India, TIJCSA Publishers & 2278-1080, Vol. 1, No 5 / pp. 1-7 / July 2012 23. Chandramowliswaran N, Srinivasan.S and Chandra Segar.T, “A Note on Linear based Set Associative Cache address System” International J. on Computer Science and Engg. (IJCSE) & India, Engineering Journals & 09753397, Vol. 4 No. 08 / pp. 1383-1386 / Aug. 2012. 24. Bellini, Emanuele, and Nadir Murru. 2015. “An Efficient and Secure RSA--like Cryptosystem Exploiting R’edei Rational Functions over Conics.” : 1–18. http://arxiv.org/abs/1511.03451. 25. “Anti-Piracy For Movies Using Forensic Water Marking”, in IJCA Digital Library on February 15, 2013, ISBN: 973-93-80872-84-0. 26. A Novel Interpolation Based Super Resolution of the Cropped Scene from a Video” Published in IJERT on March 2013, ISSN:2278-0181. 27. Uncertain Data Prediction on Dynamic Road Network, IEEE ICICES 2014. 28. Various Indexing and query processing Techniques in spatio-temporal data, ICTACT Journal on Soft Computing ( Volume: 6 , Issue: 3 ),April,2016.

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4915

Chandra Segar Thirumalai*et al. /International Journal of Pharmacy & Technology 29. M.K.Nallakaruappan M.Senthilkumar, U.Senthilkumaran, “Review of asymmetric key cryptography in wireless sensor networks,” International Journal of Engineering and Technology, Vol. 8 Issue 2 pp. 859-862, 2016 30. Dr.P.Ilango, M.Senthilkumar, “A Survey on Job Scheduling in Big Data,” in Journal of Cybernetics And Information Technologies, Vol. 16 Issue 3 pp. 35-51 31. P Viswanathan, P Venkata Krishna, “Text fusion watermarking in medical image with semi-reversible for secure transfer and authentication,” Advances in Recent Technologies in Communication and Computing, 2009. ARTCom'09. pp. 585-589 32. P Viswanathan, P Venkata Krishna, “A Joint FED Watermarking System Using Spatial Fusion for Verifying the Security Issues of Teleradiology,” IEEE Journal of Biomedical and Health Informatics, Vol.8, Issue 3, pp.753-764 33. P Viswanathan, “Fusion of cryptographic watermarking medical image system with reversible property,” in Computer Networks and Intelligent Computing 2011 pp.533-540. 34. P Viswanathan, P Venkata Krishna, “Fingerprint enhancement and compression method using Morlet wavelet,” in Inderscience Publishers 2010, Vol. 3, Issue 4, pp. 261-268 35. P Viswanathan, P Venkata Krishna,“Morlet wavelet fingerprint invariant automated authentication system,” in Int. J. of Recent Trends in Engineering and Technology, 2010, Vol. 4, Issue 1, pp. 1-5 36. Chandrasegar Thirumalai, Senthilkumar M, Vaishnavi B,“Physicians Medicament using Linear Public Key Crypto System,” in International conference on Electrical, Electronics, and Optimization Techniques

IEEE-ICEEOT,

March 2016. 37. Ellen Jochemsz, and Benne de Weger, “A Partial Key Exposure Attack on RSA using a 2-Dimensional Lattice,” LNCS, Information Security, 2006, Vol. 4176, pp.203-216 38. P Viswanathan, P Venkata Krishna, S Hariharan, “Multimodal Biometric Invariant Moment Fusion Authentication System,” in Information Processing and Management 2010, pp. 136-143 39. Chandramowliswaran, N., S. Srinivasan, and P. Muralikrishna. "Authenticated key distribution using given set of primes for secret sharing," Systems Science & Control Engineering 2015, Vol.3, Issue 1, pp. 106-112. 40. ChandrasegarThirumalai, “Physicians Drug encoding system using an Efficient and Secured Linear Public Key Cryptosystem (ESLPKC),” Vol. 8 Issue 3, Sep. 2016 pp. 16296-16303.

IJPT| Dec-2016 | Vol. 8 | Issue No.4 | 4907-4916

Page 4916