Cryptographic aspects of real hyperelliptic curves

Jacobson, Michael J.; Scheidler, Renate; Stein, Andreas

doi:10.2478/v10127-010-0030-9

Cited by 8 publications

(6 citation statements)

References 28 publications

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“…Recent results from hyperelliptic curve cryptography, particularly those from [12], employ scalar multiplication methods in which a large number of giant steps is replaced by a series of (much faster) baby steps. These could potentially be adapted to the real quadratic field infrastructure setting.…”

Section: Discussionmentioning

confidence: 99%

Improved Exponentiation and Key Agreement in the Infrastructure of a Real Quadratic Field

Dixon

Jacobson

Scheidler

2012

Progress in Cryptology – LATINCRYPT 2012

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Abstract. We describe improvements to the performance of a key agreement protocol based in the infrastructure of a real quadratic field through investigating fast methods for exponentiating ideals. We present adaptations of non-adjacent form and signed base-3 exponentiation and compare these to the binary method. To adapt these methods, we introduce new algorithms for squaring, cubing, and dividing w-near (f, p) representations of ideals in the infrastructure. Numerical results from an implementation of the key agreement protocol using our new algorithms and all three exponentiation methods are presented, demonstrating that nonadjacent form exponentiation improves the speed of key establishment for most of the currently recommended security levels.

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Section: Discussionmentioning

confidence: 99%

Improved Exponentiation and Key Agreement in the Infrastructure of a Real Quadratic Field

Dixon

Jacobson

Scheidler

2012

Progress in Cryptology – LATINCRYPT 2012

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“…In this section we recall the definition of hyperelliptic curves, the different types of hyperelliptic curves and how to pass from a real hyperelliptic curve to an imaginary hyperelliptic curve. For more detail in this section, we refer the reader to [8] [9]. A hyperelliptic curve of genus 2 g ≥ over  is defined by an equation:…”

Section: Background On Hyperelliptic Curvesmentioning

confidence: 99%

Rational Points on Genus 3 Real Hyperelliptic Curves

Miayoka¹,

Babindamana²,

Bossoto³

2021

OJDM

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We compute rational points on real hyperelliptic curves of genus 3 defined on  whose Jacobian have Mordell-Weil rank 0 r = . We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find ( ) C  . We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

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“…for some constants B 3 , B 4 , and B 5 (depending on a, but not on n). Finally, since d > 1 and m is constant, the bound in (15) implies that n is bounded. Hence Z(φ, a, b) is finite, as claimed.…”

Section: Riccati Equations In Orbits and Dynamical Zsigmondy Setsmentioning

confidence: 99%

Riccati equations and polynomial dynamics over function fields

Hindes

Jones

2019

Trans. Amer. Math. Soc.

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Given a function field K and φ ∈ K[x], we study two finiteness questions related to iteration of φ: whether all but finitely many terms of an orbit of φ must possess a primitive prime divisor, and whether the Galois groups of iterates of φ must have finite index in their natural overgroup Aut(T d ), where T d is the infinite tree of iterated preimages of 0 under φ. We focus particularly on the case where K has characteristic p, where far less is known. We resolve the first question in the affirmative under relatively weak hypotheses; interestingly, the main step in our proof is to rule out "Riccati differential equations" in backwards orbits. We then apply our result on primitive prime divisors and adapt a method of Looper to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials. We also prove that almost all quadratic polynomials over Q(t) have iterates whose Galois group is all of Aut(T d ).v(φ n (b) − a) > 0 and v(φ m (b) − a) = 0 for all 1 ≤ m ≤ n − 1.Of course, one expects that most terms in the sequence φ n (b) − a have primitive prime divisors, and to measure the failure of this heuristic, we define the Zsigmondy set of φ and the pair (a, b) to be Z(φ, a, b) := {n : φ n (b) − a has no primitive prime divisors}.Evidence suggests that Z(φ, a, b) is finite, unless the tuple (φ, a, b) is special in some way; for instance, if b has finite orbit under φ, then Z(φ, a, b) is infinite for arbitrary a ∈ K, and the same conclusion holds for a = 0 and any b in the case φ(x) = x d . If K has characterisitc zero, then Z(φ, a, b) is known to be finite in many cases [5,13,14,22], with the strongest results coming over function fields [7].On the other hand, there are very few results known for fields of positive characteristic. Nonetheless, in [9, 12] the first author was able to prove the finiteness of Zsigmondy sets for 2010 Mathematics Subject Classification: Primary: 11R32, 37P15. Secondary: 14G05.

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Cryptographic aspects of real hyperelliptic curves

Cited by 8 publications

References 28 publications

Improved Exponentiation and Key Agreement in the Infrastructure of a Real Quadratic Field

Improved Exponentiation and Key Agreement in the Infrastructure of a Real Quadratic Field

Rational Points on Genus 3 Real Hyperelliptic Curves

Riccati equations and polynomial dynamics over function fields

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