Some applications of graph spectrum in post quantum cryptography
-
R. Krishnaprabha
Abstract
Having access to an elliptic curve with a specified number of points over a specific finite field is a prerequisite for many contemporary cryptography systems. The isogenies, which are surjective morphisms, play a significant role as specific mappings between these curves. Isogeny volcanoes are graphs with elliptic curves as their vertices and l-isogenies as their edges. Two identical elliptic curves over Fq share the same trace t and cardinality. David Kohel examines the relationships between isogenies of degree l and curves in Ellt(Fq), the collection of curves defined over Fq with trace t, in his thesis. More precisely, he describes the structure of the graph of l-isogenies defined on Ellt(Fq). There is a relation between isogeny graphs and orders in OK and uses modular polynomials to find the conductor of End(E). The connected components of this graph are isogeny volcanoes and it is possible to travel through these structures using modular polynomials, even without knowing the cardinality of the curve. By using this information computation of the l-adic valuation of the trace t, for l|g possible and hence obtain some information on the cardinality of the curve. Isogeny graphs of supersingular elliptic curves plays a major role in cryptography. Isogeny based cryptography, studies cryptosystems whose security is based on the difficulty of finding a path in isogeny graphs of supersingular elliptic curves. This study investigates the use of graph adjacency matrices in maintaining the security of isogeny based key exchange protocols.
Abstract
Having access to an elliptic curve with a specified number of points over a specific finite field is a prerequisite for many contemporary cryptography systems. The isogenies, which are surjective morphisms, play a significant role as specific mappings between these curves. Isogeny volcanoes are graphs with elliptic curves as their vertices and l-isogenies as their edges. Two identical elliptic curves over Fq share the same trace t and cardinality. David Kohel examines the relationships between isogenies of degree l and curves in Ellt(Fq), the collection of curves defined over Fq with trace t, in his thesis. More precisely, he describes the structure of the graph of l-isogenies defined on Ellt(Fq). There is a relation between isogeny graphs and orders in OK and uses modular polynomials to find the conductor of End(E). The connected components of this graph are isogeny volcanoes and it is possible to travel through these structures using modular polynomials, even without knowing the cardinality of the curve. By using this information computation of the l-adic valuation of the trace t, for l|g possible and hence obtain some information on the cardinality of the curve. Isogeny graphs of supersingular elliptic curves plays a major role in cryptography. Isogeny based cryptography, studies cryptosystems whose security is based on the difficulty of finding a path in isogeny graphs of supersingular elliptic curves. This study investigates the use of graph adjacency matrices in maintaining the security of isogeny based key exchange protocols.
Chapters in this book
- Frontmatter I
- About the book VII
- Preface IX
- Contents XI
- About the Editors XV
- Logarithmic third Hankel determinant for a subclass of starlike functions 1
- Effect of gyrotactic microorganisms in rotating magnetic nanofluids in porous media 17
- Some applications of graph spectrum in post quantum cryptography 31
- Finite-time stability analysis of neutral type delay Cohen–Grossberg neural network with discontinuous activation function 47
- Influence of temperature modulation over heat/mass transport of Rivlin–Ericksen viscoelastic nanoliquid in Hele-Shaw cell 57
- On an operator preserving polynomial inequality of Paul Turán 75
- Effect of interaction between two finite faults on ground deformation 85
- On the number of bounded regions generated by a class of paths in the complex plane 111
- Binomial and de Moivre revisited through differential equations 125
- Modified projection method for second kind Fredholm integral equations with trigonometric polynomials 131
- Effect of heat and mass transfer in binary nanofluids (for silver nanoparticles) saturated in porous medium under the influence of LTNE model and different gravity modulation 145
- A comparative analysis of standard Knapsack and Legendre–Knapsack cryptosystem 169
- Study of creep in composite functionally graded disc with non-linear thickness 179
- On the h(hv)-torsion tensor and the v-curvature tensor of Matsumoto change with an h-vector 189
- Software reliability growth model considering power-law testing effort using optimization on hybrid PSO-GWO algorithm 199
- On near-perfect numbers with five prime factors 207
- Eulerian integrals associated with the product of multivariable H-function and generalized M-series 221
- Upper bounds for the complex growth rates in the Darcy–Brinkman convection in a binary viscoelastic fluid in a saturated porous layer 233
- Some result of fixed-point theorems in complete M-fuzzy metric space 243
- FR-labeling on hybrid partition shell graphs 261
- On essential M-Artinian modules 269
- Properties of reductions of multiplication modules 275
- Computational analysis of fractional vibrational equations 285
- Analysis of modified fractional order Lotka–Volterra model 307
- Topological indices of two chemical compounds used for drugs in the treatment of H1N1 335
- On the Gaussian higher-order Mersenne numbers 355
- Petri net: modeling, properties and analysis techniques 367
- An inventory system with time-dependent linear demand for deteriorating items under uncertainty with a completely backlogged 379
- On ∗-reverse derivations of involutive rings 389
- More functions via soft pre∗-generalized closed sets 397
- Modeling of piston problem for generalized Chaplygin gas 409
- A new estimation of the degree of approximation of functions belonging to Lipschitz class by Borel–Euler summability method of Fourier series 421
- Modeling the effect of discrete delay in SIR epidemic model with logistic growth 429
- Simple harmonic motion in view of quantum calculus 455
- On Bernstein and Turán-type inequalities for polynomials 461
- A weak and strong form of Ng ∗ b-continuous function in nano topological space 473
- Securing networks with graph theory: an approach to cyber security 483
Chapters in this book
- Frontmatter I
- About the book VII
- Preface IX
- Contents XI
- About the Editors XV
- Logarithmic third Hankel determinant for a subclass of starlike functions 1
- Effect of gyrotactic microorganisms in rotating magnetic nanofluids in porous media 17
- Some applications of graph spectrum in post quantum cryptography 31
- Finite-time stability analysis of neutral type delay Cohen–Grossberg neural network with discontinuous activation function 47
- Influence of temperature modulation over heat/mass transport of Rivlin–Ericksen viscoelastic nanoliquid in Hele-Shaw cell 57
- On an operator preserving polynomial inequality of Paul Turán 75
- Effect of interaction between two finite faults on ground deformation 85
- On the number of bounded regions generated by a class of paths in the complex plane 111
- Binomial and de Moivre revisited through differential equations 125
- Modified projection method for second kind Fredholm integral equations with trigonometric polynomials 131
- Effect of heat and mass transfer in binary nanofluids (for silver nanoparticles) saturated in porous medium under the influence of LTNE model and different gravity modulation 145
- A comparative analysis of standard Knapsack and Legendre–Knapsack cryptosystem 169
- Study of creep in composite functionally graded disc with non-linear thickness 179
- On the h(hv)-torsion tensor and the v-curvature tensor of Matsumoto change with an h-vector 189
- Software reliability growth model considering power-law testing effort using optimization on hybrid PSO-GWO algorithm 199
- On near-perfect numbers with five prime factors 207
- Eulerian integrals associated with the product of multivariable H-function and generalized M-series 221
- Upper bounds for the complex growth rates in the Darcy–Brinkman convection in a binary viscoelastic fluid in a saturated porous layer 233
- Some result of fixed-point theorems in complete M-fuzzy metric space 243
- FR-labeling on hybrid partition shell graphs 261
- On essential M-Artinian modules 269
- Properties of reductions of multiplication modules 275
- Computational analysis of fractional vibrational equations 285
- Analysis of modified fractional order Lotka–Volterra model 307
- Topological indices of two chemical compounds used for drugs in the treatment of H1N1 335
- On the Gaussian higher-order Mersenne numbers 355
- Petri net: modeling, properties and analysis techniques 367
- An inventory system with time-dependent linear demand for deteriorating items under uncertainty with a completely backlogged 379
- On ∗-reverse derivations of involutive rings 389
- More functions via soft pre∗-generalized closed sets 397
- Modeling of piston problem for generalized Chaplygin gas 409
- A new estimation of the degree of approximation of functions belonging to Lipschitz class by Borel–Euler summability method of Fourier series 421
- Modeling the effect of discrete delay in SIR epidemic model with logistic growth 429
- Simple harmonic motion in view of quantum calculus 455
- On Bernstein and Turán-type inequalities for polynomials 461
- A weak and strong form of Ng ∗ b-continuous function in nano topological space 473
- Securing networks with graph theory: an approach to cyber security 483
