Hybrid Authentication Cybersystem Based on Discrete Logarithm, Factorization and Array Entanglements

A hybrid cryptographic system providing digital authentication is described and analyzed in this paper. The proposed cryptosystem incorporates three features: complexity of the discrete logarithm problem, complexity of integer factorization of a product of two large primes and a combination of symmetric and asymmetric keys. In order to make the cryptosystem less vulnerable to cryptanalytic attacks a concept of digital entan-glements is introduced. As a result, the proposed cryptographic system has four layers


Introduction and Basic Definitions
In this paper a hybrid digital signature cyber-secure communication system is described and analyzed.In order to make this cryptosystem faster and less vulnerable to cryptanalytic attacks a concept of entanglements is introduced [1,2].Furthermore, in this cryptographic protocol there are four layers (entanglement-encryption-decryption-disentanglement).Since there is no one-to-one mapping between a plaintext block and the corresponding ciphertext block, this system of communication is less vulnerable to plaintext attacks.The overall cryptographic algorithm is a hybrid protocol that incorporates three features: discrete logarithm problem modulo large prime [3], factorization of a product of two large primes [4] and a combination of symmetric and asymmetric keys.
To describe the proposed cryptosystem, let's consider , ,.., r m a a a  (1) consisting of r blocks of a digitized plaintext that is to be transmitted from a sender (Alice) to a receiver (Bob); B1.A square non-singular matrix E with r r  In the paper and E are respectively called a vector and matrix of entanglements [1].C1.A sufficiently strong cryptographic protocol L that is used for encryption of one of the entanglements, for example, , with corresponding ciphertext .In order to speed up the encryption/decryption procedure and as a result to minimize the entire communication time it is necessary to minimize the amount of computations.For that reason there is no necessity to encrypt all other entanglements j i h  , where j = 1, 2,.., i -1, i + 1,.., r and is the encrypted entanglement.Indeed, if is not known to a potential intruder, then he or she must solve a system of r equations, where only r -1 components of vector h are publicly known.In the cryptosystem described below the size r of the array m is a trade-off between crypto-immunity and acceleration of the decryption: the larger the value of r, the faster the overall communication protocol.On the other hand, the larger r is, the less time is required to cryptanalyze the entire message.is a private key of the k-th user, not a public key. is a generator [5].B2.Each user selects large primes and , such that

Digital Signature Scheme
and that their product satisfies two constraints: L2.If the original array m is intelligible, but the recovered text is not, then Bob realizes that it was forged by an intruder; otherwise Bob accepts authenticity of the text.

Selection of Block Size and Matrix of Entanglements
To make sure that the entanglements are smaller than every n i , {otherwise the entire array is not recoverable}, select the matrix of entanglement E and such division of a plaintext onto blocks that the maximal value of the i-th entanglement h i does not exceed
Then (13) implies that Therefore, for every k = 1,…, r must hold p From the recovery procedure (14) it is clear that we can compute all initial blocks a, b, c, d and e only if we know all numeric values 1 2 3 4 5 from (13).Henceforth, this fact implies that it is sufficient to encrypt at least one of these entanglements to securely protect all five plaintext blocks.

, , , , h h h h h
Furthermore, it is necessary to notice that entanglements themselves do not provide secure protection.In the proposed cryptographic scheme instead of employing just one layer (plaintext/encryption/ciphertext) we propose two layers (plaintext/entanglement/encryption/ci- phertext) between the plaintext array (a, b, c, d, e,…) and ciphertext ( ,…).

, , , , c h h h h
Remark 2: The RSA algorithm discussed below is just an example of how can be encrypted.Any strong cryptocol based on the complexity of factorization of n = pq can also be used.The Rabin algorithm [6] or (hyper) elliptic-curve cryptography [7][8][9][10]  x c n  and then verifies the signature: If y is intelligible, then Bob accepts it as an authenticated message from Alice.

Examples of Entanglements
and all are integers as well.
k Proof follows from two observations:  all have the same parity which implies that their pair-wise differences are even.Therefore, every is an integer.Q. E. D.
, ,.. and r a a a Complexity of recovery: It requires r -1 subtractions and divisions by 2 (binary shifts) to recover the first r -1 blocks in (19) and r -1 subtractions to recover the last block in (20).
If a sender (Alice) encrypts only s of all entanglements, where 0 < s < r, then the intruder will not be able to deduce any blocks (provided that the matrix E is properly selected and a portion of entanglements is encrypted with a sufficiently strong PKC protocol).In an extreme case, if s = 1, then the intruder must solve a system of equations Ea = g, where the matrix E is known but only r -s elements of vector g are known.However, this is impossible, because to find the blocks the intruder must know all r elements of vector g., ,.., r a a a

Non-Linear Transformations
In the more general case, the entanglements can be non-linear, i.e. h := E(a), and/or some components of the transformation E(a) can be also encrypted.For example, if h := Ea, then we can encrypt several elements of matrix E. This approach is beyond the scope of this short paper.It is important to bear in mind that the selection of the transformation E(a) affects the computational complexity of the recovery process.
The choice of the mapping E is important.If E is a matrix, then it must be non-singular and selected in such a way that the recovery will not become a computationally formidable.
Example 3: Let's consider an array of r plaintext blocks and the following r entanglements: It is obviously sufficient to encrypt only one of the entanglements.Then, after the decryption, we proceed as follows: and for k from 2 to r - Combined with encryption these non-linear entanglements provide secure protection and recovery for every transmitted array.Yet, they require divisions of integers and extraction of square roots, which are computationally more complicated procedures.

Improper Entanglements
Therefore, r -2 blocks are cryptographically unprotected in every array.

Trade-off Analysis
Every block in ( 16)-( 17) requires four exponentiations for encryption and decryption.In contrast, in the protocol A.2-L.2 described above, ( 4)-( 11), the array of r blocks requires only one exponentiation for its encryption and decryption.Therefore, the larger the transmitted array r is, the more efficient the speed-up of A.2-L.2 is.If r = 100, then A.2-L.2 is four hundred times faster than the RSA algorithm.
Furtermore, if B A n m n   , then the RSA digital signature algorithm ( 16)-( 17) fails to recover the original plaintext m unless special measures are taken [4,11].The application of entanglements (linear or non-linear transformations) is a tool that is proposed to accelerate the encryption-decryption process.Although the entanglements themselves do not provide protection, yet, when used in combination with other measures, they decrease the amount of computations necessary for the entire encryption/decryption process.It is necessary to mention that any detailed and credible quantification of the trade-off between the size r of the array and cryptoimmunity requires analysis of all strategies potentially available to the intruder.Yet to qualitatively illustrate this point of view, let's consider an asymptotic case, where the size r of the transmitted array of plaintext blocks is very large.From one point of view, the larger r is, the more advantageous the proposed cryptosystem is.Indeed, only one entanglement is encrypted/decrypted instead of all r entanglements as it is done in the RSA, ElGamal, Rabin [6], ECC [7][8][9] and other PKC cryptosystems [10].On the other hand, if the size r of the array is very large, then the intruder can invest the required time and computing resources to cryptanalyze the encrypted entanglement.
Let's consider an extreme case, where the entire message M consists of N blocks.Let's select a square nonsingular N × N matrix E, compute N entanglements 1 2

, ,.., N h h h
1 h using ( 18) and encrypt only one of them, say, .For instance, if the sender transmits information re-garding highly-sensitive issues of long-term national policy or the details of a major corporate policy, the intruder will invest all available resources to break the encrypted entanglement [12][13][14][15][16][17][18].Therefore, for security purposes, it is safer to divide the entire file M onto several parts/arrays and securely protect each array.

Decryption: Reduction of Complexity
The most serious computational bottleneck of the present public-key cryptographic protocols is that they are notoriously slow and therefore cannot be used in the real-time exchange of sensitive information.
Although we are far away from completely eliminating this bottleneck, the proposed cryptosystem is a systemic tool that accelerates secure communication via open channels of the Internet or within corporate networks.
Eliminating the bottleneck mentioned above is one of major research areas today and will likely occupy hundreds of communication specialists and system designers for years ahead.Various PKC algorithms were introduced in the last thirty years.Elliptic-curve cryptography and its hyper-elliptic extension are vivid examples of this research: to accelerate the encryption/decryption process.The proposed cryptosystem is another illustration of how we can accelerate the PKC protocols if the entangled arrays rather than individual blocks are encrypted.

Illustrative Numeric Example
The steps A4-H4 describe a system design stage and the steps I4-L4 describe its implementation for signed encryption and authenticated decryption of arrays p q p q and   where p q p q , (30) and compute their products [1]:

Conclusions
This paper describes a novel concept for the PKC that employs a combination of DLP, factorization and entanglements, which facilitates otherwise computationally difficult problem [12,14,19].Let's summarize the most important issues that were described and briefly discussed in this paper: A: In contrast with RSA, k is a private key of the k-th user, not the public key [19]; n B: In another contrast, the encryption/decryption is applied not to every block of the plaintext, but to every array of blocks; in other words, the unit of protection is not a block, but an array of several blocks [20]; C: Within each array prior to encryption all blocks are entangled [1]; D: The advantage of entanglements is that they are interdependent; the disadvantage is that if one entanglement is corrupted, it affects the entire array.Namely, that array cannot be recovered by the receiver [2]; E: If the information is transmitted in an aggressive media and subject to networking failures or errors, the proposed cryptosystem cannot be used unless additional measures of information assurance are applied (see [21,22]).
F: As a by-product of interdependence, there is no necessity to encrypt and decrypt each block or each entanglement.Instead it is sufficient to encrypt only one of r entanglements [23].This is the first advantage of the proposed protocol.
G: The application of cryptography based on a cubic-root provides the second advantage.The encryption requires only two multiplications [1]; H: The overhead of the entanglements is on the stage of information recovery: it is necessary to solve a system of r equations with r unknowns.Yet, there are many ways how to select matrix E that will make these computations easier.Several linear and non-linear examples of entanglements are provided above for illustration.Additional examples of entanglements are described in [20].The proposed cryptosystem also provides a digital signature protocol.

1 h 1 c
To avoid confusions, it is important to indicate the following distinctions: The matrix of entanglement E (and non-linear mappings) discussed below are not secret keys as in an affine cryptographic algorithm; all elements of matrix E are publicly known; In contrast with the RSA and Rabin algorithms, k n 2 2

4. Essence of RSA Digital Signature Algorithm
based on modulo composite n are other possible applications.
i h 2.k c : mod B d B