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dieter schmidt

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University of Zagreb

Oxford Brookes University

Cruz Vargas-De-León

Dr. J. M. Ashfaque (MInstP)

Independent Researcher

New Mexico State University

Florentin Smarandache

University of New Mexico

The University Of Jordan

Yildiz Technical University

Muğla Sıtkı Koçman üniversitesi

Tomasz Szemberg

University of the National Education Commission in Krakow

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Papers by dieter schmidt

Rainbow, a New Multivariable Polynomial Signature Scheme

Applied Cryptography and Network Security, 2005

Balanced Oil and Vinegar signature schemes and the unbalanced Oil and Vinegar signature schemes a... more Balanced Oil and Vinegar signature schemes and the unbalanced Oil and Vinegar signature schemes are public key signature schemes based on multivariable polynomials. In this paper, we suggest a new signature scheme, which is a generalization of the Oil-Vinegar construction to improve the efficiency of the unbalanced Oil and Vinegar signature scheme. The basic idea can be described as a construction of multi-layer Oil-Vinegar construction and its generalization. We call our system a Rainbow signature scheme. We propose and implement a practical scheme, which works better than Sflash v 2 , in particular, in terms of signature generating time.

Cryptanalysis of HFEv and Internal Perturbation of HFE

Lecture Notes in Computer Science, 2005

Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis a... more Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis and Shamir showed that to make the cryptosystem secure, a special parameter D of any HFE cryptosystem can not be too small. Consequently Kipnis, Patarin and Goubin proposed an enhanced variant of the HFE cryptosystem by combining the idea of Oil and Vinegar construction with the HFE construction. Essentially they "perturb" the HFE system with some external variables. In this paper, we will first present a new cryptanalysis method for the HFEv schemes. We then use the idea of internal perturbation to build a new cryptosystem, an internally perturbed HFE cryptosystem (IPHFE).

Bifurcations of relative equilibria in the 4- and 5-body problem

Ergodic Theory and Dynamical Systems, 1988

The equilateral triangle family of relative equilibria of the 4-body problem consists of three pa... more The equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary massmat the centroid. For one value of the massmthis relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family asmpasses through the degenerate value.The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary massmat the centroid. For one value of the massmthis relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family asmpasses through the degenerate value.

Algebraic Attack on HFE Revisited

Lecture Notes in Computer Science

Complexity Estimates for the F 4 Attack on the Perturbed Matsumoto-Imai Cryptosystem

Cryptography and Coding, 2005

Though the Perturbed Matsumoto-Imai (PMI) cryptosystem is considered insecure due to the recent d... more Though the Perturbed Matsumoto-Imai (PMI) cryptosystem is considered insecure due to the recent differential attack of Fouque, Granboulan, and Stern, even more recently Ding and Gower showed that PMI can be repaired with the Plus (+) method of externally adding as few as 10 randomly chosen quadratic polynomials. Since relatively few extra polynomials are added, the attack complexity of a Gröbner basis attack on PMI+ will be roughly equal to that of PMI. Using Magma's implementation of the F4 Gröbner basis algorithm, we attack PMI with parameters q = 2, 0 ≤ r ≤ 10, and 14 ≤ n ≤ 59. Here, q is the number of field elements, n the number of equations/variables, and r the perturbation dimension. Based on our experimental results, we give estimates for the running time for such an attack. We use these estimates to judge the security of some proposed schemes, and we suggest more efficient schemes. In particular, we estimate that an attack using F4 against the parameters q = 2, r = 5, n = 96 (suggested in [7]) has a time complexity of less than 2 50 3-DES computations, which would be considered insecure for practical applications.

The Family of Direct Periodic Orbits of the First Kind in the Restricted Problem of Three Bodies

Ordinary Differential Equations, 1972

Publisher Summary This chapter discusses the family ofdirect periodic orbits of the first kind in... more Publisher Summary This chapter discusses the family ofdirect periodic orbits of the first kind in the restricted problem of three bodies. For small mass ratios μ, the family of direct periodic orbits of the first kind becomes with increasing h near a = (1/2)2/3 a family of the second kind, which terminates by triplication on an orbit in the family of retrograde periodic orbits of the first kind. The planar restricted problem of three bodies is defined in barycentric, synodic coordinates by the Hamiltonian function. For μ = 0, the problem reduces to the central force problem in a rotating coordinate system. A standard theorem asserts that if the nontrivial characteristic multipliers are not +1, then for small values of μ the circular orbit can be continued to an orbit of the restricted problem, which Poincare calls of the first kind.

The New Implementation Schemes of the TTM Cryptosystem Are Not Secure

Coding, Cryptography and Combinatorics, 2004

We show that the new TTM implementation schemes have a defect. There exist linearization equation... more

Bifurcations of Relative Equilibria in the N-Body and Kirchhoff Problems

SIAM Journal on Mathematical Analysis, 1988

The bifurcations of a one-parameter family of relative equilibria in the N-body problem are studi... more The bifurcations of a one-parameter family of relative equilibria in the N-body problem are studied using normal form theory, Lie transforms, and an algebraic processor. The one-parameter family consists of N-1 bodies of mass at the vertices of a regular polygon and one body of mass m at the centroid. As N increases there are more and more values of the mass parameter m where the relative equilibrium is degenerate. For N =< 13 each of these degenerates gives rise to a bifurcation and a new relative equilibrium. This is established using a computer-aided proof. A similar analysis is carried out for the N-vortex problem of Kirchhoff.

Normalization of the Hamiltonian Near L 4 for Small Values of μ

Qualitative Theory of Dynamical Systems, 2012

Periodic orbits in the circular restricted problem of three bodies near L 4 have been studied ext... more

Hopf's Bifurcation Theorem and the Center Theorem of Liapunov with resonance cases

Journal of Mathematical Analysis and Applications, 1978

Growth of the Ideal Generated by a Quadratic Multivariate Function Over GF(3)

Journal of Algebra and Its Applications, 2013

Let K be the field GF(3). We calculate the growth of the ideal Aλ where A is the algebra of funct... more Let K be the field GF(3). We calculate the growth of the ideal Aλ where A is the algebra of functions from Kn → Kn and λ is a quadratic function. Specifically we calculate dim Akλ where Ak is the space of polynomials of degree less than or equal to k. This question arises in the analysis of the complexity of Gröbner basis attacks on multivariate quadratic cryptosystems such as the Hidden Field Equation systems. We also prove analogous results over the associated graded ring [Formula: see text] and state conjectures for the case of a general finite field of odd order.

Librations of central configurations and braided Saturn rings

Celestial Mechanics & Dynamical Astronomy, 1993

ABSTRACT We give a simple mathematical model for braided rings of a planet based on Maxwell&#... more

The main problem of lunar theory solved by the method of Brown

Celestial Mechanics, 1982

Brown's method for solving the main problem of lunar theory has been adapted for the computation ... more Brown's method for solving the main problem of lunar theory has been adapted for the computation by machine with the help of an algebraic processor. Brown's results are first recovered and refined. The solution is then expanded to include most terms of order nine. The terms in the series for the longitude and latitude are listed with an accuracy of 0.000 01" and of 0.000 001" for the parallax.

The lunar theory of Hill and Brown

Celestial Mechanics, 1980

Hill and Brown solved the equations of motions of the Moon with the help of multiple Fourier seri... more

Zhuang-zi: A new algorithm for solving multivariate polynomial equations over a finite field

We present the Zhuang-Zi algorithm, a new method for solving multivariate polynomial equations ov... more

Elliptic relative equilibria in the N-body problem

Journal of Differential Equations, 2005

A planar central configuration of the N-body problem gives rise to a solution where each particle... more A planar central configuration of the N-body problem gives rise to a solution where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homothety motion. If the Keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium. The totality of such solutions forms a four-dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement. In our coordinate system, the linear variational equations of such a solution decouple into three subsystems. One subsystem simply gives the motion of the center of mass, another is Kepler's problem and the third determines the nontrivial characteristic multipliers. Using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three-body problem. We reproduce the analytic studies of G. Roberts. We also study the linear stability of the four-and five-body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or square and the remaining body is at the center with arbitrary mass .

Rainbow, a New Multivariable Polynomial Signature Scheme

Applied Cryptography and Network Security, 2005

Balanced Oil and Vinegar signature schemes and the unbalanced Oil and Vinegar signature schemes a... more Balanced Oil and Vinegar signature schemes and the unbalanced Oil and Vinegar signature schemes are public key signature schemes based on multivariable polynomials. In this paper, we suggest a new signature scheme, which is a generalization of the Oil-Vinegar construction to improve the efficiency of the unbalanced Oil and Vinegar signature scheme. The basic idea can be described as a construction of multi-layer Oil-Vinegar construction and its generalization. We call our system a Rainbow signature scheme. We propose and implement a practical scheme, which works better than Sflash v 2 , in particular, in terms of signature generating time.

Cryptanalysis of HFEv and Internal Perturbation of HFE

Lecture Notes in Computer Science, 2005

Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis a... more Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis and Shamir showed that to make the cryptosystem secure, a special parameter D of any HFE cryptosystem can not be too small. Consequently Kipnis, Patarin and Goubin proposed an enhanced variant of the HFE cryptosystem by combining the idea of Oil and Vinegar construction with the HFE construction. Essentially they "perturb" the HFE system with some external variables. In this paper, we will first present a new cryptanalysis method for the HFEv schemes. We then use the idea of internal perturbation to build a new cryptosystem, an internally perturbed HFE cryptosystem (IPHFE).

Bifurcations of relative equilibria in the 4- and 5-body problem

Ergodic Theory and Dynamical Systems, 1988

The equilateral triangle family of relative equilibria of the 4-body problem consists of three pa... more The equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary massmat the centroid. For one value of the massmthis relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family asmpasses through the degenerate value.The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary massmat the centroid. For one value of the massmthis relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family asmpasses through the degenerate value.

Algebraic Attack on HFE Revisited

Lecture Notes in Computer Science

Complexity Estimates for the F 4 Attack on the Perturbed Matsumoto-Imai Cryptosystem

Cryptography and Coding, 2005

Though the Perturbed Matsumoto-Imai (PMI) cryptosystem is considered insecure due to the recent d... more Though the Perturbed Matsumoto-Imai (PMI) cryptosystem is considered insecure due to the recent differential attack of Fouque, Granboulan, and Stern, even more recently Ding and Gower showed that PMI can be repaired with the Plus (+) method of externally adding as few as 10 randomly chosen quadratic polynomials. Since relatively few extra polynomials are added, the attack complexity of a Gröbner basis attack on PMI+ will be roughly equal to that of PMI. Using Magma's implementation of the F4 Gröbner basis algorithm, we attack PMI with parameters q = 2, 0 ≤ r ≤ 10, and 14 ≤ n ≤ 59. Here, q is the number of field elements, n the number of equations/variables, and r the perturbation dimension. Based on our experimental results, we give estimates for the running time for such an attack. We use these estimates to judge the security of some proposed schemes, and we suggest more efficient schemes. In particular, we estimate that an attack using F4 against the parameters q = 2, r = 5, n = 96 (suggested in [7]) has a time complexity of less than 2 50 3-DES computations, which would be considered insecure for practical applications.

The Family of Direct Periodic Orbits of the First Kind in the Restricted Problem of Three Bodies

Ordinary Differential Equations, 1972

Publisher Summary This chapter discusses the family ofdirect periodic orbits of the first kind in... more Publisher Summary This chapter discusses the family ofdirect periodic orbits of the first kind in the restricted problem of three bodies. For small mass ratios μ, the family of direct periodic orbits of the first kind becomes with increasing h near a = (1/2)2/3 a family of the second kind, which terminates by triplication on an orbit in the family of retrograde periodic orbits of the first kind. The planar restricted problem of three bodies is defined in barycentric, synodic coordinates by the Hamiltonian function. For μ = 0, the problem reduces to the central force problem in a rotating coordinate system. A standard theorem asserts that if the nontrivial characteristic multipliers are not +1, then for small values of μ the circular orbit can be continued to an orbit of the restricted problem, which Poincare calls of the first kind.

The New Implementation Schemes of the TTM Cryptosystem Are Not Secure

Coding, Cryptography and Combinatorics, 2004

We show that the new TTM implementation schemes have a defect. There exist linearization equation... more

Bifurcations of Relative Equilibria in the N-Body and Kirchhoff Problems

SIAM Journal on Mathematical Analysis, 1988

The bifurcations of a one-parameter family of relative equilibria in the N-body problem are studi... more The bifurcations of a one-parameter family of relative equilibria in the N-body problem are studied using normal form theory, Lie transforms, and an algebraic processor. The one-parameter family consists of N-1 bodies of mass at the vertices of a regular polygon and one body of mass m at the centroid. As N increases there are more and more values of the mass parameter m where the relative equilibrium is degenerate. For N =< 13 each of these degenerates gives rise to a bifurcation and a new relative equilibrium. This is established using a computer-aided proof. A similar analysis is carried out for the N-vortex problem of Kirchhoff.

Normalization of the Hamiltonian Near L 4 for Small Values of μ

Qualitative Theory of Dynamical Systems, 2012

Periodic orbits in the circular restricted problem of three bodies near L 4 have been studied ext... more

Hopf's Bifurcation Theorem and the Center Theorem of Liapunov with resonance cases

Journal of Mathematical Analysis and Applications, 1978

Growth of the Ideal Generated by a Quadratic Multivariate Function Over GF(3)

Journal of Algebra and Its Applications, 2013

Let K be the field GF(3). We calculate the growth of the ideal Aλ where A is the algebra of funct... more Let K be the field GF(3). We calculate the growth of the ideal Aλ where A is the algebra of functions from Kn → Kn and λ is a quadratic function. Specifically we calculate dim Akλ where Ak is the space of polynomials of degree less than or equal to k. This question arises in the analysis of the complexity of Gröbner basis attacks on multivariate quadratic cryptosystems such as the Hidden Field Equation systems. We also prove analogous results over the associated graded ring [Formula: see text] and state conjectures for the case of a general finite field of odd order.

Librations of central configurations and braided Saturn rings

Celestial Mechanics & Dynamical Astronomy, 1993

ABSTRACT We give a simple mathematical model for braided rings of a planet based on Maxwell&#... more

The main problem of lunar theory solved by the method of Brown

Celestial Mechanics, 1982

Brown's method for solving the main problem of lunar theory has been adapted for the computation ... more Brown's method for solving the main problem of lunar theory has been adapted for the computation by machine with the help of an algebraic processor. Brown's results are first recovered and refined. The solution is then expanded to include most terms of order nine. The terms in the series for the longitude and latitude are listed with an accuracy of 0.000 01" and of 0.000 001" for the parallax.

The lunar theory of Hill and Brown

Celestial Mechanics, 1980

Hill and Brown solved the equations of motions of the Moon with the help of multiple Fourier seri... more

Zhuang-zi: A new algorithm for solving multivariate polynomial equations over a finite field

We present the Zhuang-Zi algorithm, a new method for solving multivariate polynomial equations ov... more

Elliptic relative equilibria in the N-body problem

Journal of Differential Equations, 2005

A planar central configuration of the N-body problem gives rise to a solution where each particle... more A planar central configuration of the N-body problem gives rise to a solution where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homothety motion. If the Keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium. The totality of such solutions forms a four-dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement. In our coordinate system, the linear variational equations of such a solution decouple into three subsystems. One subsystem simply gives the motion of the center of mass, another is Kepler's problem and the third determines the nontrivial characteristic multipliers. Using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three-body problem. We reproduce the analytic studies of G. Roberts. We also study the linear stability of the four-and five-body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or square and the remaining body is at the center with arbitrary mass .