39 research outputs found
Discrete Logarithms in Generalized Jacobians
D\'ech\`ene has proposed generalized Jacobians as a source of groups for
public-key cryptosystems based on the hardness of the Discrete Logarithm
Problem (DLP). Her specific proposal gives rise to a group isomorphic to the
semidirect product of an elliptic curve and a multiplicative group of a finite
field. We explain why her proposal has no advantages over simply taking the
direct product of groups. We then argue that generalized Jacobians offer poorer
security and efficiency than standard Jacobians
On the existence of distortion maps on ordinary elliptic curves
Distortion maps allow one to solve the Decision Diffie-Hellman problem on
subgroups of points on the elliptic curve. In the case of ordinary elliptic
curves over finite fields, it is known that in most cases there are no
distortion maps. In this article we characterize the existence of distortion
maps in the remaining cases.Comment: 3 Pages (Updated version corrects an error in the previous version
SYMMETRIC BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE ALGEBRA
Elliptic Curve Cryptography (ECC) has gained widespread adoption in the field of cryptography due to its efficiency and security properties. Symmetric bilinear pairings on elliptic curves have emerged as a powerful tool in cryptographic protocols, enabling advanced constructions and functionalities. This paper explores the intersection of symmetric bilinear pairings, elliptic curves, and Lie algebras in the context of cryptography. We provide a comprehensive overview of the theoretical foundations, applications, and security considerations of this amalgamation
ΠΡΠΈΠ½ΡΠΈΠΏΡ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΠΊΡΠΈΠΏΡΠΎΠ³ΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ Π°Π»Π³ΠΎΡΠΈΡΠΌΠΎΠ² Π½Π° ΡΠ»Π»ΠΈΠΏΡΠΈΡΠ΅ΡΠΊΠΈΡ ΠΊΡΠΈΠ²ΡΡ
ΠΠ°Π²Π΅Π΄Π΅Π½ΠΎ ΠΏΡΠΈΠ½ΡΠΈΠΏΠΈ ΠΏΠΎΠ±ΡΠ΄ΠΎΠ²ΠΈ ΠΊΡΠΈΠΏΡΠΎΠ³ΡΠ°ΡΡΡΠ½ΠΈΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΡΠ² Π½Π° Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΈΡ
ΠΊΡΠΈΠ²ΠΈΡ
, Π²ΠΈΠΊΠΎΡΠΈΡΡΠ°Π½Ρ Π² ΡΡΠ°Π½Π΄Π°ΡΡΡ ΠΠ‘Π’Π£ 4145-2002, ΡΠΊΡ Π·Π°Π±Π΅Π·ΠΏΠ΅ΡΡΡΡΡ Π²ΠΈΡΠΎΠΊΡ ΠΊΡΠΈΠΏΡΠΎΠ³ΡΠ°ΡΡΡΠ½Ρ ΡΡΡΠΉΠΊΡΡΡΡ ΡΠΈΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΡΠ΄ΠΏΠΈΡΡ.Design principles of elliptic curve cryptographic algorithms are explained. It is shown that implementation of these principles in the National digital signature standard DSTU 4145-2002 guarantees a high cryptographic strength of the digital signature
Efficient hash maps to G2 on BLS curves
When a pairing e:G1ΓG2βGT, on an elliptic curve E defined over a finite field Fq, is exploited for an identity-based protocol, there is often the need to hash binary strings into G1 and G2. Traditionally, if E admits a twist E~ of order d, then G1=E(Fq)β©E[r], where r is a prime integer, and G2=E~(Fqk/d)β©E~[r], where k is the embedding degree of E w.r.t. r. The standard approach for hashing into G2 is to map to a general point PβE~(Fqk/d) and then multiply it by the cofactor c=#E~(Fqk/d)/r. Usually, the multiplication by c is computationally expensive. In order to speed up such a computation, two different methodsβby Scott et al. (International conference on pairing-based cryptography. Springer, Berlin, pp 102β113, 2009) and by Fuentes-Castaneda et al. (International workshop on selected areas in cryptography)βhave been proposed. In this paper we consider these two methods for BLS pairing-friendly curves having kβ{12,24,30,42,48}, providing efficiency comparisons. When k=42,48, the application of Fuentes et al. method requires expensive computations which were infeasible for the computational power at our disposal. For these cases, we propose hashing maps that we obtained following Fuentes et al. idea.publishedVersio