379 research outputs found
PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES
A new general mathematical problem, suitable for public-key cryptosystems, is proposed: morphism computation in a category of Abelian groups. In connection with elliptic curves over finite fields, the problem becomes the following: compute an isogeny (an algebraic homomorphism) between the elliptic curves given. The problem seems to be hard for solving with a quantum computer. ElGamal public-key encryption and Diffie-Hellman key agreement are proposed for an isogeny cryptosystem. The paper describes theoretical background and a public-key encryption technique, followed by security analysis and consideration of cryptosystem parameters selection. A demonstrative example of encryption is included as well
Isogenies and cryptography
This thesis explores the notion of isogenies and its applications to cryptography. Elliptic curve cryptography (ECC) is an efficient public cryptosystem with a short key size. For this reason it is suitable for implementing on memory-constraint devices such as smart cards, mobile devices, etc. However, these devices leak information about their private key through side channels (power consumption, electromagnetic radiation, timing etc) during cryptographic processing. In this thesis we have examined countermeasures against a specific side channel attack (power consumption) using isogeny, (a rational homomorphism between elliptic curves) and elliptic curve isomorphism. We found that these methods are an efficient way of securing cryptographic devices using ECC against power analysis attacks. We have also investigated the security and efficiency of implementation of a public key cryptosystem based on isogenies. We found that in order to implement the proposed cryptosystem one has to compute a root of the Hilbert polynomial H D ( X ) over F p . Since there is no known efficient way of achieving this calculation, the proposed cryptosystem cannot be used in practice
Isogeny-based post-quantum key exchange protocols
The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented
Isogenies and Cryptography
This thesis explores the notion of isogenies and its applications to cryptography. Elliptic curve cryptography (ECC) is an efficient public cryptosystem with a short key size. For this reason it is suitable for implementing on memory-constraint devices such as smart cards, mobile devices, etc. However, these devices leak information about their private key through side channels (power consumption, electromagnetic radiation, timing etc) during cryptographic processing. In this thesis we have examined countermeasures against a specific side channel attack (power consumption) using isogeny, (a rational homomorphism between elliptic curves) and elliptic curve isomorphism. We found that these methods are an efficient way of securing cryptographic devices using ECC against power analysis attacks. We have also investigated the security and efficiency of implementation of a public key cryptosystem based on isogenies. We found that in order to implement the proposed cryptosystem one has to compute a root of the Hilbert polynomial HD(X) over Fp. Since there is no known efficient way of achieving this calculation, the proposed cryptosystem cannot be used in prac
Algorithms and cryptographic protocols using elliptic curves
En els darrers anys, la criptografia amb corbes el.líptiques ha
adquirit una importància creixent, fins a arribar a formar part en
la actualitat de diferents estàndards industrials. Tot i que s'han
dissenyat variants amb corbes el.líptiques de criptosistemes
clàssics, com el RSA, el seu màxim interès rau en la seva
aplicació en criptosistemes basats en el Problema del Logaritme
Discret, com els de tipus ElGamal. En aquest cas, els
criptosistemes el.líptics garanteixen la mateixa seguretat que els
construïts sobre el grup multiplicatiu d'un cos finit primer, però
amb longituds de clau molt menor.
Mostrarem, doncs, les bones propietats d'aquests criptosistemes,
així com els requeriments bàsics per a que una corba
sigui criptogràficament útil, estretament relacionat amb la seva
cardinalitat. Revisarem alguns mètodes que permetin descartar
corbes no criptogràficament útils, així com altres que permetin
obtenir corbes bones a partir d'una de donada. Finalment,
descriurem algunes aplicacions, com són el seu ús en Targes
Intel.ligents i sistemes RFID, per concloure amb alguns avenços
recents en aquest camp.The relevance of elliptic curve cryptography has grown in recent
years, and today represents a cornerstone in many industrial
standards. Although elliptic curve variants of classical
cryptosystems such as RSA exist, the full potential of elliptic
curve cryptography is displayed in cryptosystems based on the
Discrete Logarithm Problem, such as ElGamal. For these, elliptic
curve cryptosystems guarantee the same security levels as their
finite field analogues, with the additional advantage of using
significantly smaller key sizes.
In this report we show the positive properties of elliptic curve
cryptosystems, and the requirements a curve must meet to be
useful in this context, closely related to the number of points.
We survey methods to discard cryptographically uninteresting
curves as well as methods to obtain other useful curves from
a given one. We then describe some real world applications
such as Smart Cards and RFID systems and conclude with a
snapshot of recent developments in the field
A Survey Report On Elliptic Curve Cryptography
The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP. Keywords: Elliptic curve cryptography (ECC), isogenies, hyperelliptic curve (HEC) , Discrete Logarithm Problem (DLP), Integer Factorization , Binary Field, Prime FieldDOI:http://dx.doi.org/10.11591/ijece.v1i2.8
OSIDH and SiGamal : cryptosystems from supersingular elliptic curves (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)
We introduce two cryptosystems, OSIDH and SiGamal, which use isogenies between supersingular elliptic curves over a finite field. And we consider computational problems on which these cryptosystems are based. In particular, we discuss a relation between these problems and problems to find the image of a point under a secret isogeny
Post-Quantum Cryptography from Supersingular Isogenies (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)
This paper is based on a presentation made at RIMS conference on “Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties”, so-called “Supersingular 2020”. Post-quantum cryptography is a next-generation public-key cryptosystem that resistant to cryptoanalysis by both classical and quantum computers. Isogenies between supersingular elliptic curves present one promising candidate, which is called isogeny-based cryptography. In this paper, we give an introduction to two isogeny-based key exchange protocols, SIDH [17] and CSIDH [2], which are considered as a standard in the subject so far. Moreover, we explain briefly our recent result [24] about cycles in the isogeny graphs used in some parameters of SIKE, which is a key encapsulation mechanism based on SIDH
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