## Cryptography through Interpolation, Approximation and Computational Intelligence Methods (2003)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Meletiou03cryptographythrough,

author = {G. C. Meletiou and D. K. Tasoulis and M. N. Vrahatis},

title = {Cryptography through Interpolation, Approximation and Computational Intelligence Methods},

year = {2003}

}

### OpenURL

### Abstract

Recently, numerous techniques and methods have been proposed to address hard and complex algebraic and number theoretical problems related to cryptography.

### Citations

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Citation Context ...ude that the network has learned the problem “satisfactorily”. The total number of epochs required can be considered as the speed of the method. More sophisticated training techniques can be found in =-=[12, 14, 17, 18, 20, 38, 45]-=-. 4.1. Experimental Results The numerical experiments performed address the discrete logarithm problem in modular arithmetic, and the factorization problem, through neural networks. The empirical test... |

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Citation Context ...geometry, etc. Cryptosystems rely on the assumption that these problems are computationally intractable, in the sense that their computation cannot be completed in polynomial time, e.g. factorization =-=[39]-=-, discrete logarithm [1, 32, 33, 37], knapsack problem [24]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, int... |

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Citation Context ... The empirical tests were performed using a C++ neural network interface built under Linux operating system with the gcc compiler. The training methods considered were: Standard Back Propagation (BP) =-=[40]-=-, Back Propagation with Variable Stepsize (BPVS) [18], Resilient Back Propagation (RPROP) [38] and On-Line Adaptive Back Propagation (OABP) [17]. All the methods were extensively tested with a wide ra... |

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Citation Context ...q must be computationally intractable. The cryptanalyst has to compute a from b. To achieve this it is sufficient to obtain φ(N) from N, or equivalently to factorize N, since φ(N) = (p − 1) · (q − 1) =-=[23, 39]-=-. 3. Interpolation and Approximation Methods In a finite field Fq every function can be represented as a polynomial (Lagrangian interpolation). For every function f : Fq → Fq there exists a unique pol... |

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Citation Context ...ropagated to the following layers until the final outputs of the network are computed. The computational power of neural networks derives from their inherent ability to adapt to specific problems. In =-=[13, 47]-=- the following statement has been proved: “Standard feedforward networks with only a single hidden layer can approximate any continuous function uniformly on any compact set and any measurable functio... |

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Citation Context ...or real polynomials. The lower n 5 ln n bounds 0.14 ln n and have been computed for the degree and the sparsity of the polynomials, respectively. Finally, we would like to point out here that Shor in =-=[43]-=- has shown that polynomial algorithms for factorization and index computation exist for quantum computers. Provided that quantum computing becomes an available technology in the future, RSA and discre... |

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Citation Context ...ude that the network has learned the problem “satisfactorily”. The total number of epochs required can be considered as the speed of the method. More sophisticated training techniques can be found in =-=[12, 14, 17, 18, 20, 38, 45]-=-. 4.1. Experimental Results The numerical experiments performed address the discrete logarithm problem in modular arithmetic, and the factorization problem, through neural networks. The empirical test... |

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Citation Context ...minimizing the total error function then it is obvious that its aim has been fulfilled. Thus training is a nontrivial minimization problem. The most popular training method is back propagation method =-=[12]-=-, which is based on the well-known steepest descent method. The back propagation learning process applies small iterative steps which correspond to the training epochs. At each epoch t the method upda... |

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Citation Context ...f an integer z satisfying the following relation: g z = h, where g is a primitive element of a finite field, and h a non–zero element. The security of various public–key and private–key cryptosystems =-=[1, 5, 9, 26, 27, 29, 30, 33, 32, 37, 48, 49]-=- is based on the assumption that DLP is computationally intractable. More specifically, we refer to: (1) the Diffie–Hellman exchange protocol [6, 32], (2) the El Gamal public key cryptosystem as well ... |

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Citation Context ...ems rely on the assumption that these problems are computationally intractable, in the sense that their computation cannot be completed in polynomial time, e.g. factorization [39], discrete logarithm =-=[1, 32, 33, 37]-=-, knapsack problem [24]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic a... |

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Citation Context ...ure correspondence we intent to apply various other networks and learning techniques such as Learning Rate Adaptation methods [20, 45], Non-Monotone neural networks [2], Probabilistic neural networks =-=[44]-=-, Self-Organized Map algorithm [14], Recurrent networks and Radial Basis function networks [12]. In conclusion our experience indicates that the Neural Network approach on problems related to cryptogr... |

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Citation Context ...ems rely on the assumption that these problems are computationally intractable, in the sense that their computation cannot be completed in polynomial time, e.g. factorization [39], discrete logarithm =-=[1, 32, 33, 37]-=-, knapsack problem [24]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic a... |

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Citation Context ...ese problems are computationally intractable, in the sense that their computation cannot be completed in polynomial time, e.g. factorization [39], discrete logarithm [1, 32, 33, 37], knapsack problem =-=[24]-=-. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic algorithms (e.g., see [1... |

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Citation Context ...ropagated to the following layers until the final outputs of the network are computed. The computational power of neural networks derives from their inherent ability to adapt to specific problems. In =-=[13, 47]-=- the following statement has been proved: “Standard feedforward networks with only a single hidden layer can approximate any continuous function uniformly on any compact set and any measurable functio... |

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Citation Context ...ems rely on the assumption that these problems are computationally intractable, in the sense that their computation cannot be completed in polynomial time, e.g. factorization [39], discrete logarithm =-=[1, 32, 33, 37]-=-, knapsack problem [24]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic a... |

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Citation Context ...4]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic algorithms (e.g., see =-=[1, 5, 21, 27, 29, 30, 32, 33, 37, 48]-=-.62 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis In this paper we focus on the interpolation and approximation techniques [3, 4, 5, 10, 16, 22, 25, 26, 27, 29, 30, 31, 41, 42, 48, 49]. More specific... |

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Citation Context ...es, and generic algorithms (e.g., see [1, 5, 21, 27, 29, 30, 32, 33, 37, 48].62 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis In this paper we focus on the interpolation and approximation techniques =-=[3, 4, 5, 10, 16, 22, 25, 26, 27, 29, 30, 31, 41, 42, 48, 49]-=-. More specifically, we consider polynomial interpolation, discrete Fourier transforms and finally, polynomial approximation. In a sense Artificial Neural Networks (ANNs) can be considered as generali... |

29 |
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Citation Context ...4]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic algorithms (e.g., see =-=[1, 5, 21, 27, 29, 30, 32, 33, 37, 48]-=-.62 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis In this paper we focus on the interpolation and approximation techniques [3, 4, 5, 10, 16, 22, 25, 26, 27, 29, 30, 31, 41, 42, 48, 49]. More specific... |

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Citation Context ...es, and generic algorithms (e.g., see [1, 5, 21, 27, 29, 30, 32, 33, 37, 48].62 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis In this paper we focus on the interpolation and approximation techniques =-=[3, 4, 5, 10, 16, 22, 25, 26, 27, 29, 30, 31, 41, 42, 48, 49]-=-. More specifically, we consider polynomial interpolation, discrete Fourier transforms and finally, polynomial approximation. In a sense Artificial Neural Networks (ANNs) can be considered as generali... |

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Citation Context ...es, and generic algorithms (e.g., see [1, 5, 21, 27, 29, 30, 32, 33, 37, 48].62 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis In this paper we focus on the interpolation and approximation techniques =-=[3, 4, 5, 10, 16, 22, 25, 26, 27, 29, 30, 31, 41, 42, 48, 49]-=-. More specifically, we consider polynomial interpolation, discrete Fourier transforms and finally, polynomial approximation. In a sense Artificial Neural Networks (ANNs) can be considered as generali... |

12 |
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Citation Context ...4]. Numerous techniques have been proposed to address these problems including algebraic methods, number theory, software oriented methods, interpolation techniques, and generic algorithms (e.g., see =-=[1, 5, 21, 27, 29, 30, 32, 33, 37, 48]-=-.62 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis In this paper we focus on the interpolation and approximation techniques [3, 4, 5, 10, 16, 22, 25, 26, 27, 29, 30, 31, 41, 42, 48, 49]. More specific... |

8 |
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Citation Context ...d decryption) functions are defined as functions over finite fields, so it is natural to try to express them as polynomials. Regarding the discrete logarithm function, the well–known formula of Wells =-=[46]-=- exists: log a(x) = ∑p−2 i=1 xi , 1 − ai ( x = 0, a, x ∈ Zp, a is a generator of the Z ∗) p . The above mentioned formula can be generalized in the case of a field of prime power order Fq and in the ... |

6 |
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Citation Context ...nction is indeed a computationally hard function. Another approach is to consider the notion of linear complexity in order to measure the complexity of a problem. We recall that the linear complexity =-=[7, 8, 15, 22]-=- of a sequence {si} is the smallest positive integer m such that there are constants66 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis c1, c2, . . . , cm satisfying the equation: si = c1 · si−1 + c2 · ... |

6 | Explicit form for the discrete logarithm over the field GF(p
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6 | On polynomial representations of Boolean functions related to some number theoretic problems
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Citation Context ...decision problems. The Boolean functions permit representations as multivariate polynomials [42]. Interesting results related to the factorization problem inspired from cryptography can been found in =-=[36]-=-. In particular, lower bounds for the degree and the sparsity of polynomials representing the Boolean function, which decides whether a given n-bit integer is square free, have been found. The represe... |

5 |
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4 | Pseudonoise Sequences
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Citation Context ...nction is indeed a computationally hard function. Another approach is to consider the notion of linear complexity in order to measure the complexity of a problem. We recall that the linear complexity =-=[7, 8, 15, 22]-=- of a sequence {si} is the smallest positive integer m such that there are constants66 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis c1, c2, . . . , cm satisfying the equation: si = c1 · si−1 + c2 · ... |

4 |
Vrahatis M.N. and Androulakis G.S., Effective backpropagation training with variable stepsize
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Citation Context ...ude that the network has learned the problem “satisfactorily”. The total number of epochs required can be considered as the speed of the method. More sophisticated training techniques can be found in =-=[12, 14, 17, 18, 20, 38, 45]-=-. 4.1. Experimental Results The numerical experiments performed address the discrete logarithm problem in modular arithmetic, and the factorization problem, through neural networks. The empirical test... |

4 |
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3 | Polynomial representations of the DiffieHellman mapping
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3 |
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3 |
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Citation Context ...d networks with only a single hidden layer can approximate any continuous function uniformly on any compact set and any measurable function to any desired degree of accuracy”. It has also been proved =-=[35]-=- that a single hidden layer feedforward network with r units in the hidden layer, has a lower bound on the degree of the approximation of any function. The lower bound obstacle can be alleviated if mo... |

2 | Linear complexity of the discrete logarithm
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Citation Context ...nction is indeed a computationally hard function. Another approach is to consider the notion of linear complexity in order to measure the complexity of a problem. We recall that the linear complexity =-=[7, 8, 15, 22]-=- of a sequence {si} is the smallest positive integer m such that there are constants66 G.C. Meletiou, D.K. Tasoulis, & M.N. Vrahatis c1, c2, . . . , cm satisfying the equation: si = c1 · si−1 + c2 · ... |

2 |
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1 |
Vrahatis M.N., Artificial nonmonotonic neural networks
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Citation Context ...edforward neural networks. In a future correspondence we intent to apply various other networks and learning techniques such as Learning Rate Adaptation methods [20, 45], Non-Monotone neural networks =-=[2]-=-, Probabilistic neural networks [44], Self-Organized Map algorithm [14], Recurrent networks and Radial Basis function networks [12]. In conclusion our experience indicates that the Neural Network appr... |

1 |
Vrahatis M.N., Adaptive stepsize algorithms for on-line training of neural networks
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1 |
Androulakis G.S., On the alleviation of the problem of local minima
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Citation Context ...t all training methods yielded suboptimal solutions, which can be attributed to the local minima effect. To alleviate convergence to local minima we applied the recently proposed deflection technique =-=[19]-=- and the function “stretching” method [34]. A large variety of network topologies were considered. As expected, different topologies exhibited great differences in the results. After extensive experim... |

1 |
Androulakis G.S., Increasing the convergence rate of the error backpropagation algorithm by learning rate adaptation methods
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1 |
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1 |
Vrahatis M.N., A first study of the neural network approach to the RSA cryptosystem
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Citation Context ...A. Our experimental results conform to this conclusion. To the best of our knowledge, this is the first attempt to apply neural networks to tackle difficult problems related to Cryptography (see also =-=[28]-=-). Our experience indicates that it is possible to train feedforward neural networks to address this very difficult problem. In particular for small prime numbers we have shown that it is possible to ... |

1 |
Plagianakos V.P., Magoulas G.D. and Vrahatis M.N., Objective function “stretching” to alleviate convergence to local minima, Nonlinear Analysis
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Citation Context ...solutions, which can be attributed to the local minima effect. To alleviate convergence to local minima we applied the recently proposed deflection technique [19] and the function “stretching” method =-=[34]-=-. A large variety of network topologies were considered. As expected, different topologies exhibited great differences in the results. After extensive experimentation,Cryptography through Interpolati... |

1 |
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