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ANALYZING THE STRENGTH OF PELL’S RSA
Chandrasegar T1, Senthilkumar M1, R.Silambarasan2, Carlos Becker Westphall 3
1Assistant Professor (Senior), SITE school, VIT University, India
2M.Tech Student ,SITE School, VIT University, Vellore, India.
3Professor, Federal University of Santa Catarina, Departamento de Informática e Estatística
Florianópolis, Santa Catarina, Brazil.
By taking two distinct Diophantine equations with distinct large prime numbers and two different co-ordinates, two
secret keys are generated at the same time so that two users can encrypt their messages / plaintext. Hence two sets of
public key exponents with one set of private key exponents were the result of proposed algorithm.
Keywords: Pell’s equation, Diophantine equation, Public key cryptosystem.
The traditional RSA asymmetric key cryptographic system dates to be first in the public key cryptography comprising
public key creation, encrypting original message, private key generation and decrypting the encrypted message. In this
system the selection of prime numbers pairs larger and larger ensures the strong protection to cipher text from the third
parties in cracking back to original plaintext with means of different attacking schemes in the existing literature
including continued fraction method private key retrieval, Euclidean and extended Euclidean division algorithm method
of tracking private key are some with few. Continued fraction constraint of private key value less than that of modular
value to the power constant is admissible only when the two distinct prime numbers are limited to certain bit length.
When this bit length increased the retrieval process with continued fraction method would lead to longer time for
computing. Same scenario holds true for extended Euclidean division method as well. Next with RSA cryptography
method only one user can make encryption (or) in other lines, user’s single message only able to encrypt at the time.
Albeit holds this fine as far as the asymmetric key cryptography, the time and cost computing raised the issue which
tended further study in this regime. As an instance either of public key generation (or) private key generation cost only
can be made less also in time, by making large variations to one another. There hence give rise the study of re-balanced
RSA where encryption key and decryption key are balanced with respect to the aspects of time and cost followed by the
Chinese Remainder Theorem method of private key tracing from RSA and led to RSA-CRT Likewise in the different
direction first method of dual key generation by taking small public key exponent at one time and small private key
exponent on the other led to Dual RSA key generation where the results are provisioned with Lattice based breaking
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system and analyzed to re-balanced RSA and RSA-CRT. The similar kind of producing two public keys by the simple
linear equations such as Diophantine equation what under influence called Pell’s equation is explained in this work. This
algorithm does not four sets of equations and two sets of co-ordinates system as in Dual RSA. Rather by simply taking
two Diophantine equations the two sets of secret keys are produced thence two users at a time can encrypt their message.
The proposed algorithm is proved with the help of numerical example calculations.
In  taking four straight lines (linear equations) with two co-ordinates are comprised to give four set of prime numbers
name yp 1 ;q 1 ;p 2 ;q 2 and iterated until q 2 satisfies for the prime numbers albeit other three too need to be prime.
From these four prime numbers where x0 sandy 0s are chosen in such a way that two instances of public and private keys
were the result. Exactness of the key generation in either case were verified through Dual RSA equations form its key. In
 trailing the method of Euclidean algorithm, modified trial division method is employed in the private key generation
which back from the known modulus value and public key as prime numbers be-come larger extent this method helped
in encrypting digital data’s.
In  the encryption key generation is explained in the means of prime pairs selected for Euler’s totient function
calculation followed by system modulus computation. This leads to public key generation study in terms of prime
numbers and primitive roots. In  Knapsack problem with Merkle-Hellman number theoretic concepts are employed in
public key exponent calculation and showed the hardness of breaking back of private key. Defined their strong ability of
scheme against various known attacks including brute force. In Prime number selection procedure in public key from
modulus of system and totient function of Euler computation are further extended and studied and proved the proposed
system secure against Shamir attacks. In  library function in C++ comprising encryption key generation and
groupware technique in encapsulation method described in encrypting and decrypting files stored in windows platform.
In  n carry array is used in the calculation of public key when the prime numbers are larger and proved the efficiency
in digital signature along with class lib in C++ were given. In  Pell’s equation first introduced in RSA cryptography in
key generation and management by fixing threshold value and proved there scheme is strong against coalition attack.
In  secret key generation from Pell’s equation by taking the roots of Diophantine equation for the constant prime is
proposed and analyzed its complexity with N-prime RSA, Dual RSA dn traditional RSA. Crypt analysis of Fermat’s
attack, Weiner’s continued fraction and extended Euclidean method were compared along with numerical examples. In
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thick communication, two Diophantine equations are taken at a time and showed two secret key sets along with private
key exponent set. Proposed system/ algorithm makes two different users to encrypt their messages and send while
receiver receives two cipher texts and decrypt to original text with single private key set, thus the proposed scheme
makes the communication to one receiver to two senders at a time. While in other go, the sender can encrypt their two
different messages with two secret keys and communicate with receiver and receiver performs as explained earlier. The
proposed algorithm is explained with numerical example to prove its efficiency.
Prof Pell studied the Diophantine equation of the form y2 dx 2 = 1. For the given positive values d the x and y values
should be found which satisfying the governing equation. One such solution is ford= 61, x and y has values
respectively2261590and1776319041.In this work the secret key generation is based on the above mentioned Diophantine
but he d values are chosen as prime number in due course positive prime integers. The proposed algorithm takes two
Diophantine equations and followed by selection of two distinct prime integers for d and then values of x0s and y0s
which satisfying Pell’s equation. Detailed proposed system is given in algorithm 1.
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Therefore output of algorithm 1 produces two set of public keys are obtained namely(
1;d;P;X1;N)and(2;d;Q;X2;N) with the private key(e3;N).Continuing in same way Dual RSA is further expanded to
generate one more pair set of instance in key, the following algorithm variant to Dual RSA is proposed and takes the
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Hence the output of algorithm 2 is(e;N1 =p 1 q 1 ;n 2 = p 2 q 2 ;n 3 = p 3 q 3) and (d;p1;q1;p2;q2;p3;q3). Also it was
found that algorithm 2 satisfies the three simultaneous equation so called key equations for variation of Dual RSA key
Proposed system algorithm 1 is verified by taking numerical examples such as with respect to first Diophantine equation
when P = 7the prime number the values satisfying are X1= 3and Y1= 8. For the second Diophantine equation when
prime number Q= 11satisfies for X2= 3and Y2= 10where the first three steps are cover. From step 4 of algorithm 1
taking a = 13and b = 17givesN = 221and -(N) = 192. From step 5 the e value is taken as e= 5. From Eqs (1) and (2) of
algorithm 1 calculations yields 1= 39552and 2= 40100 which satisfies Eqs (3) and (4) respectively. From step 9 the d
values found to be d= 77. From Eqs (5) and (6) values of s1 and s2 are calculated and yielding s1= 149and s2 = 149.
Considering the plaintext M= 19, step11 gives the cipher text C= 15and in turn step 12 retracts original message M=
19from Eqs (7) and (8) respectively. Therefore two set of public keys are (39552;77;7;3;221) and (40100;77;11;3;221).
Along with the private key set(53;221).Example of proposed algorithm 2 takes, for x1= 20 and x2= 9if step 1 of
algorithm 2 gives p1 =181 which is prime number. For y2 = 12 is step 2 yields p2= 241which is prime. For y1= 8 in step
3 gives p3 = 73which is prime. From step 4 q1= 97which is prime number. Step 5 takes e = 7and d = 12343where k1=
2is taken satisfying step 5. Then from steps 6 and 7 the values are q2= 19and q3= 41both are prime. Therefore output of
algorithm 2 is (7;17557;4579;2993) and (12343;181;97;241;19;73;41). From this example it can be verified that three
key equations are satisfied with above values similar to that of Dual RSA two key equations.
Upon this communication two algorithms are pro-posed in key generating scheme for RSA crypto-graphic system. While
the Pell’s RSA key generation produces single pair of key, an variant in aspects of two sets production in key generation
by considering two distinct Pell’s equations the strength is further improved to one step. Presented numerical example
proves the efficiency of designed system also makes two users to encrypt their data. It can be further expanded by
considering simultaneous Pell’s system of equations in key generating where n users able to cipher their data
Alternate to the Dual RSA scheme of key generation, further k values were considered and designed the so called trivial
RSA one step ahead to Dual RSA where three instances of RSA are generated which improves the security than that of
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Dual RSA. In the pipeline working of variant of Dual RSA the trivial RSA algorithm 2 is explained and proved with
numerical example. Also believed which can be further developed by taking n number of k 0 s and hence multiple
instances of RSA well be deployed as algorithm 2.
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