Quasi-quadratic elliptic curve point counting using rigid cohomology

We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the canonical lift of E, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.Comment: 14 page

Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations

Ideas from Fourier analysis have been used in cryptography for the last three decades. Akavia, Goldwasser and Safra unified some of these ideas to give a complete algorithm that finds significant Fourier coefficients of functions on any finite abelian group. Their algorithm stimulated a lot of interest in the cryptography community, especially in the context of `bit security'. This manuscript attempts to be a friendly and comprehensive guide to the tools and results in this field. The intended readership is cryptographers who have heard about these tools and seek an understanding of their mechanics and their usefulness and limitations. A compact overview of the algorithm is presented with emphasis on the ideas behind it. We show how these ideas can be extended to a `modulus-switching' variant of the algorithm. We survey some applications of this algorithm, and explain that several results should be taken in the right context. In particular, we point out that some of the most important bit security problems are still open. Our original contributions include: a discussion of the limitations on the usefulness of these tools; an answer to an open question about the modular inversion hidden number problem

An Improved Public Key Cryptography Based on the Elliiptic Curve

Elliptic curve cryptography offers two major benefits over RSA: more security per bit, and a suitable key size for hardware and modern communication. Thus, this results to smaller size of public key certificates, lower power requirements and smaller hardware processors. Three major approaches are used in this dissertation to enhance the elliptic curve cryptsystems: reducing the number of the elliptic curve group arithmetic operations, speeding up the underlying finite field operations and reducing the size of the transited parameters. A new addition formula in the projective coordinate is introduced, where the analysis for this formula shows that the number of multiplications over the finite field is reduced to nine general field element multiplications. Thus this reduction will speed up the computation of adding two points on the elliptic curve by 11 percent. Moreover, the new formula can be used more efficiently when it is combined with the suggested sparse elements algorithms. To speed up the underlying finite field operations, several new algorithms are introduced namely: selecting random sparse elements algorithm, finding sparse base points, sparse multiplication over polynomial basis, and sparse multiplication over normal basis. The complexity analysis shows that whenever the sparse techniques are used, the improvement rises to 33 percent compared to the standard projective coordinate formula and improvement of 38 percent compared to affine coordinate. A new algorithm to compress and decompress the sparse elements algorithms are introduced to reduce the size of the transited parameters. The enhancements are applied on three protocols and two applications. The protocols are Diffie-Hellman, ELGamal and elliptic curve digital signature. In these protocols the speed of encrypting, decrypting and signing the message are increased by 23 to 38 percent. Meanwhile, the size of the public keys are reduced by 37 to 48 percent. The improved algorithms are applied to the on-line and off-line electronic payments systems, which lead to probably the best solution to reduce the objects size and enhance the performance in both systems

A Generic Approach to Searching for Jacobians

We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3} with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio

Agri-Food Traceability Management using a RFID System with Privacy Protection

In this paper an agri-food traceability system based on public key cryptography and Radio Frequency Identification (RFID) technology is proposed. In order to guarantee safety in food, an efficient tracking and tracing system is required. RFID devices allow recording all useful information for traceability directly on the commodity. The security issues are discussed and two different methods based on public cryptography are proposed and evaluated. The first algorithm uses a nested RSA based structure to improve security, while the second also provides authenticity of data. An experimental analysis demonstrated that the proposed system is well suitable on PDAs to

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

Hard isogeny problems over RSA moduli and groups with infeasible inversion

We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of cryptanalytic attempts on these problems. Moreover, based on the hardness of these problems, we provide a construction of groups with infeasible inversion, where the underlying groups are the ideal class groups of imaginary quadratic orders. Recall that in a group with infeasible inversion, computing the inverse of a group element is required to be hard, while performing the group operation is easy. Motivated by the potential cryptographic application of building a directed transitive signature scheme, the search for a group with infeasible inversion was initiated in the theses of Hohenberger and Molnar (2003). Later it was also shown to provide a broadcast encryption scheme by Irrer et al. (2004). However, to date the only case of a group with infeasible inversion is implied by the much stronger primitive of self-bilinear map constructed by Yamakawa et al. (2014) based on the hardness of factoring and indistinguishability obfuscation (iO). Our construction gives a candidate without using iO.Comment: Significant revision of the article previously titled "A Candidate Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the constructions by giving toy examples, added "The Parallelogram Attack" (Sec 5.3.2). 54 pages, 8 figure

A Comparison of Cryptography Courses

The author taught two courses on cryptography, one at Duke University aimed at non-mathematics majors and one at Rose-Hulman Institute of Technology aimed at mathematics and computer science majors. Both tried to incorporate technical and societal aspects of cryptography, with varying emphases. This paper will discuss the strengths and weaknesses of both courses and compare the differences in the author's approach.Comment: 14 pages; to appear in Cryptologi