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Optimal Strategies for Computation of Degree degree ℓⁿ Isogenies for SIDH

100%

Wroński M. , Chojnacki A.

tom Vol. 66, No. 3

465--472

This article presents methods and algorithms for the computation of isogenies of degree ℓⁿ. Some of these methods are obtained using recurrence equations and generating functions. A standard multiplication based algorithm for computation of isogeny of degree ℓⁿ has time complexity equal to O(n²M (n log n)), where M (N) denotes the cost of integers of size N multiplication. The memory complexity of this algorithm is equal to O (n log (n log (n))). In this article are presented algorithms for: - determination of optimal strategy for computation of degree ℓⁿ isogeny, - determination of cost of optimal strategy of computation of ℓⁿ isogeny using solutions of recurrence equations, - determination of cost of optimal strategy of computation of ℓⁿ isogeny using recurrence equations, where optimality in this context means that, for the given parameters, no other strategy exists that requires fewer operations for computation of isogeny. Also this article presents a method using generating functions for obtaining the solutions of sequences (սₘ) and (cₘ) where cₘ denotes the cost of computations of isogeny of degree ℓᵘᵐum for given costs p, q of ℓ-isogeny computation and ℓ-isogeny evaluation. These solutions are also used in the construction of the algorithms presented in this article.

Vibration analysis and modelling of light-weight robot arms

100%

Leniowski R. , Wroński M.

tom Vol. 33, nr 2

art. no. 2022220

Lightweight robots (LWR) are a new generation of devices intended to be used not only for industrial tasks but also to perform actions in the human environment. This work presents an analysis of selected basic problems related to the vibration properties of light-weight robot arms. The study of vibration is based on the analysis of the root locus on the plane of complex variables. It turns out that their distribution is non-stationary and depends on the parameters of the model (arm geometry, material parameters), but also depends on the type of realised motion, which is not so obvious. Depending on the manoeuvres conducted (acceleration / deceleration), the system may lose (or increase) its oscillating properties at higher frequencies, as well as introduce a structural (measurable) delay. Recognition of the discussed properties along with their modelling is an important element of the design process of the control system of modern, light-weight robots.

High-degree compression functions on alternative models of elliptic curves and their applications

80%

Wroński M. , Kijko T. , Dryło R.

tom Vol. 184, nr 2

107--139

This paper presents method for obtaining high-degree compression functions using natural symmetries in a given model of an elliptic curve. Such symmetries may be found using symmetry of involution [–1] and symmetry of translation morphism τ T = P + T , where T is the n -torsion point which naturally belongs to the E (𝕂) for a given elliptic curve model. We will study alternative models of elliptic curves with points of order 2 and 4, and specifically Huff’s curves and the Hessian family of elliptic curves (like Hessian, twisted Hessian and generalized Hessian curves) with a point of order 3. We bring up some known compression functions on those models and present new ones as well. For (almost) every presented compression function, differential addition and point doubling formulas are shown. As in the case of high-degree compression functions manual investigation of differential addition and doubling formulas is very difficult, we came up with a Magma program which relies on the Gröbner basis. We prove that if for a model E of an elliptic curve exists an isomorphism φ : E → E M , where E M is the Montgomery curve and for any P ∈ E (𝕂) holds that φ (P ) = (φ x (P ), φ y (P )), then for a model E one may find compression function of degree 2. Moreover, one may find, defined for this compression function, differential addition and doubling formulas of the same efficiency as Montgomery’s. However, it seems that for the family of elliptic curves having a natural point of order 3, compression functions of the same efficiency do not exist.