A mathematical model for ascertaining same ciphertext generated from distinct plaintext in Michael O. Rabin Cryptosystem

Michael O. Rabin Cryptosystem can generate same ciphertext form different plaintext as well as multiple plaintext from single cyphertext. There are a number of techniues to reveal original plaintext. But none of them can seperate same cyphertext against each plaintext generated from modular reduction arithmetic. If question arises about how one can distinguish particular ciphertext against each plaintext, to answer those questions, I design a new mathematical model for identifying same ciphertext against each plaintext and it also facilitates message encryption and decryption. The proposed mathematical model constructiond based on quadratic root of quadratic residue, quadratic quotient, floor function and absolute value counting in order to identify the ciphertext against the plaintext. In particular. When same number of residues generated from multiple plaintext applying modular reduction arithmetic. The proposed crypto intensive technique uses symmetric key using Diffie-Hellman key exchange protocol. The advantage of proposed crypto intensive technique is intended receiver getting only one plainvalue distinguishing the ciphertext against the plaintext. The proposed crypto teachnique requires less time complexity and probabily secure against man-in-the-middle, chosen plaintext and cyphertext attack.


INTRODUCTION
ince [1][2] publication on January (1976,1979) by Michael O. Rabin, a huge number of surveys had been carried out over Rabin's Cryptosystem to find out its efficiency and devise a new method for real life application. It was the first asymmetric cryptosystem in the field of public key Cryptography. Security of Rabin's encryption mechanism relies on prime integer factorization. It was not widely used due to having some computational error, but its theoretical significance widespread. The encryption mechanism used quadratic residue to produce cipher text and Decryption was accomplished by Computing two square root, Bezout's Coefficient using extended Euclidean algorithm and combining them with Chinese Remainder theorem. Similarly to the RSA and ElGamal cryptosystems, Michael O. Rabin cryptosystem is described in a ring under addition and multiplication modulo composite integer. One of the main disadvantage is to generate four results during decryption and extra effort needed to sourt out the right one out of four possibilities. In this paper I design a new crypto intensive technique based on Diffie -Hellman key exchange protocol [3], concept of square modular arithmetic from Michael O. Rabin Cryptosystem, Floor function and absolute value function. The symmetric key generates from Diffie-Hellman key exchange protocol. The sender Bob sends a pair of integer to Alice as an encrypted text (C) = (└ m 2 / K ┘, m 2 mod K). After receiving, Alice decrypts the message (D) = │� . + │ and gets only one desired plain text unlike Rabin's Cryptosystem in which she gets four different decryption results. The rest of the paper is organized as a follows. Section 1.1 summarizes Overview of Michael O. Rabin cryptosystem. Ssection 1.2 gives an overview of Rabin's Signature Scheme, Section 1.3 provides an overview of Diffie-Hellman Key Exchange protocol. Section 2 gives Literature Review, Section 3.for proposed mathetical model, Section 3.1 for prposed Algorithm, Section 3.2 gives summary of proposed mathmetical model, In section 3.3 shows comparisons, Finally, Section 4, 5 give conclusion and acknowledgement.

Overview of Rabin's Cryptosystem [4]
SUMMARY: Each entity creates a public key and a corresponding private key. Each entity A should do the following: 1. Generate two large random (and distinct) primes p and q, each roughly the same size. 2. Compute n = p q. 3. A's public key is n; A's private key is (p, q).
Algorithm for Rabin's public-key encryption SUMMARY: B encrypts a message m for A, which decrypts. To recover plaintext m from c, A should find the four square roots m 1 , m 2 , m 3 , and m 4 of c modulo n. The message sent was either m 1 , m 2 , m 3 , or m 4 . A decides which of these is m by ascertain replicating bits.

Encryption:
Suppose that the last six bits of original messages are required to be replicated prior to encryption. In order to encrypt the 10bit message m = 1001111001, B replicates the last six bits of m to obtain the 16-bit message m = 1001111001111001, which in decimal notation is m = 40569. B then computes c = m 2 mod n = 405692 mod 91687 = 62111 and sends this to A.

Overview of Rabin's Signature Scheme
Rabin's Cryptosystem is composed of Key Setup, Encryption and Decryption. Key Generation step-1: Let, Alice chooses two random prime numbers P and Q. Compute public key N= P*Q she also picks a random integer (0 ≤ b < N; publicize (N, b) as her public key material, and keep (P and Q) as her private key . Encryption step-2: The sender Bob creates cipher text C=m (m +b) mod N. Here uses of b is Security purpose only (0 ≤ b < N).

Decryption step-3: Alice solves the quadratic equation m 2 -m b + c ≡≡0 (mod N)
to decrypt the ciphertext. Decryption involves computing square roots modulo N. Decryption consisting of m 2 ≡a (mod n). This is performed by solving M p = m 2 ≡a (mod p) and M q = m 2 ≡a (mod q). Pick a random integer b in range 0….p and compute the Legendre symbol (b 2 -4a) /p i.e., (b 2 -4a) (P -1) / 2 mod p with result p -1 replaced by -1, until that's -1.Now setup the second degree polynomial arithmetic f and then compute the polynomial x (p + 1)/2 mod f and x (q + 1)/2 mod f using polynomial arithmetic modulo the polynomial f. Compute Bezout's coefficient using extended Euclidean algorithm and combine these using the Chinese Remainder Theorem yielding four solutions in most cases, and picking the right one in some way. Example: Step 1. Let, two random prime number p= 41, q= 53 and public key: N= p. q =1273 Message m=92. Cipher text c =m 2 mod N = 1945. Now compute M p = m 2 ≡ a (mod p) =18 and M q = m 2 ≡ a (mod q) =37.
Step 2. Choose a random b = 2 satisfying the condition and setup a polynomial f = x 2 -b. x + M p with coefficients in Z 41 , that is f = x 2 + 39x + 18 similarly b = 4 satisfying the condition and setup a polynomial f = x 2 + 49x + 37 with coefficients in Z 53; x is the variable of the polynomial and has no particular value.
Step 3. Compute the polynomial x (p+1)/2 mod f= x 21 mod f. The binary representation of the exponential order (21) is 10101, and compute x 2 , x 4 , x 5 , x 10 , x 20 and finally x 21 mod f by left-to-right binary exponentiation.
Computation of x 20 mod f that is (10x+39) 2 mod f that is 37x+8. Computation of x 21 mod f that is 37x 2 +8x mod f. Finally, the x term has surprised leaving 31. Thus m 2 ≡a (mod p) has solution M ∈ {10, 31} (mod p).
Step 4. Compute the polynomial x (q+1)/2 mod f that is x 27 mod f using polynomial arithmetic modulo the polynomial f. The binary representation of the exponential order (27) is 11011, and compute x 2 , x 3 , x 6 , x 12 , x 13 , x 26 and finally x 27 mod f by left-toright binary exponentiation. Similar computation of step 3. Solve m 2 ≡a (mod q), with solution M ∈ {14, 39} (mod q).

Diffie-Hellman Key Exchange protocol [5]
The first published public-key algorithm appeared in the seminal paper by Diffie and Hellman that defined public-key cryptography [8]. It is generally referred to as Diffie-Hellman key exchange protocol. A number of commercial products employ this key exchange technique. The purpose of the algorithm is to enable two users to securely exchange a key that can then be used for subsequent encryption and decryption of messages. The algorithm itself is limited to the exchange of secret values. The Diffie-Hellman algorithm depends for its effectiveness on the difficulty of computing discrete logarithms

Global Public elements
q is a prime number which can define a domain so called curve area or elliptic curve, α is a primitive root of q such thatα α < q.

Literature Review
There are many surveys have been dedicated over Rabin's cryptosystem. Recently various modifications of Rabin's cryptosystem have been published in different scientific journals.
Hayder Raheem Hashim [6] proposed an update methodology that used three private keys instead of two. Consequently, the eight non-deterministic plaintext generates from one cypher text. One of them is real plaintext. The advantage of this technique is to make confusing attacker while it is very annoying to receiver as extra effort is required to distinguish original plaintext out of eight text.
Yahia Awad et al. [7] proposed a deterministic method depending on the domain of Gaussian Integer to select right plaintext among four decryption result. Recipient can decide particular plain text form four possible decryption result by selecting obtained square root with redundancies in its imaginary part (a + bi). This is the main benefit of using Gaussian integer technique. The disadvantage, on the other hand, same cyphertext can be generated from different plaintext due to having modular reduction arithmetic. For example, for the four plaintext (m) = {13, 20, 57, 64}, the same cipher text c=15.
Manish Bhatt et al. [8] extended a deterministic technique adding duplicating bits at the beginning of plaintext before encryption. Added replicating bits reflected within one decrypted text among four possible plaintext. The annoying thing is other three false result that refers to time complexity and memory complicity.
Masahiro Kaminaga, et al,. [9] discussed a fault attack technique on modular exponentiation during Rabin's encryption where a complicated situation arose in case of message reconstruction when message and public key were not relatively prime. They also provided a rigorous algorithm to handle message reconstruction.
Haytham Gani [10] performed study over Rabin and RSA Cryptosystem and provided insightful discussion. The computation speed of RSA and Rabin's Cryptosystem were roughly same. Both algorithm's security relied on prime integer factorization.
Preeti Chandrakar [11] discussed about a secure two factor remote authentication scheme using Rabin Cryptosystem. This paper showed an extended usages of Rabin's cryptosystem.

Summary of Proposed mathmetical Model
The proposed crypto technique ensures secure communication among two parties. For example, at the initial stage Alice and Bob create a shared secret key. In the second stage Bob choose a message A = 065 according to ASCII -Binary Character

Comparisons
The comparison between proposed crypto technique and Michael O. Rabin Cryptosystem as follows.

Rabin's Crypto Scheme Proosed Crypto technique
Cyphertext is a quadratic residue.
Ciphertext is a pair of integer

Conclusion
The proposed crypto intensive mathematical technique is efficient for solving four to one mapping ciphertext. Its objective to identify each cipher text separately because modular arithmetic can generate same cyphertext from different plaintext. The proposed model can efficiently identify each cipher text separately generated form modular reduction arithmetic, while Rabin's cryptosystem fails. There is a security vulnerability in symmetric key geration stage that is man in the middle attack because it does not authenticate the participants. Even thouth proposed scheme ensures security as computation procedure is unknown to adversary.

Acknowledgement
I am very grateful to my family members who supported financially to conduct study because without their financial support, love and affection, this work could not be carried out. . I thank Md. Maruf Hassan for his inpiratiional advice and Dr. Md. Mostafijur Rahman (Assistant professor, Department of software Engineering, Daffodil International University) for insightful discussion during the preparation of this paper. This work is a part of academic curriculum fulfillment for MSc in software engineering.