Arithmetic Aspects of Orbifold Pairs

Concretely efficient interactive oracle proofs (IOPs) are of interest due to their applications to scaling blockchains, their minimal security assumptions, and their potential future-proof resistance to quantum attacks.Scalable IOPs, in which prover time scales quasilinearly with the computation size and verifier time scales poly-logarithmically with it, have been known to exist thus far only over a set of finite fields of negligible density, namely, over “FFT-friendly” fields that contain a sub-group of size \(2^\mathsf{{k}} \).Our main result is to show that scalable IOPs can be constructed over any sufficiently large finite field, of size that is at least quadratic in the length of computation whose integrity is proved by the IOP. This result has practical applications as well, because it reduces the proving and verification complexity of cryptographic statements that are naturally stated over pre-defined finite fields which are not “FFT-friendly”. Prior state-of-the-art scalable IOPs relied heavily on arithmetization via univariate polynomials and Reed–Solomon codes over FFT-friendly fields. To prove our main result and extend scalability to all large finite fields, we generalize the prior techniques and use new algebraic geometry codes evaluated on sub-groups of elliptic curves (elliptic curve codes). We also show a new arithmetization scheme that uses the rich and well-understood group structure of elliptic curves to reduce statements of computational integrity to other statements about the proximity of functions evaluated on the elliptic curve to the new family of elliptic curve codes.

We classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.

We construct complex surfaces with genus two fibrations over \({\mathbb P^1}\) having special fibres such that the minimum of the multiplicities of the components is ≥ 2 whereas the g.c.d is 1. We can then produce new examples of fibred surfaces without multiple fibres which are of “general type” according to the definition of Campana. We prove that these surfaces are of general type and simply connected; and we compute in some cases their invariants. Moreover, we extend the construction obtaining general type fibrations of any even genus on simply connected surfaces. All our examples are defined over number fields.

Let E / Q E/\mathbb {Q} be an elliptic curve and let Q ( 3 ∞ ) \mathbb {Q}(3^\infty ) be the compositum of all cubic extensions of Q \mathbb {Q} . In this article we show that the torsion subgroup of E ( Q ( 3 ∞ ) ) E(\mathbb {Q}(3^\infty )) is finite and we determine 20 possibilities for its structure, along with a complete description of the Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves, and a complete list of j j -invariants for each of the 4 that do not.

The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve – which we refer to as "inter-universal Teichmüller theory" – by applying the theory of semi-graphs of anabelioids , Frobenioids , the étale theta function , and log-shells developed in earlier papers by the author. We begin by fixing what we call "initial \Theta -data" , which consists of an elliptic curve E_F over a number field F , and a prime number l\ge 5 , as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite étale coverings to the once-punctured elliptic curve X_F determined by E_F . These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring \mathbb{F}_l=\mathbb{Z}/l\mathbb{Z} acting on the l -torsion points of the elliptic curve. We then construct " \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters" associated to the given \Theta -data. These \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field – which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field – are, in some sense, "dismantled" or "disentangled" from one another. All \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters are isomorphic to one another, but may also be related to one another by means of a " \Theta -link" , which relates certain Frobenioid-theoretic portions of one \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater to another in a fashion that is not compatible with the respective conventional ring/scheme theory structures . In particular, it is a highly nontrivial problem to relate the ring structures on either side of the \Theta -link to one another. This will be achieved, up to certain "relatively mild indeterminacies" , in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an "alien ring structure" [associated, say, to the domain of the \Theta -link] in terms of a given ring structure [associated, say, to the codomain of the \Theta -link] will be applied in the final paper of the series to obtain results in diophantine geometry . Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p -adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

In this paper, we give examples of elliptic curves E/K over a number field K satisfying the property that there exist P1, P2 K[t] such that the twists and are of positive rank over K(t). As a consequence of this result on twists, we show that for those elliptic curves E/K, and for each , the rank of E over the fixed field (Kab) under is infinite, where Kab is the maximal abelian extension of K.

The equations considered in this chapter are in essence different both from the Thue equation and from its direct generalisations. It is possible to prove the existence of an effective bound for solutions of these equations by purely arithmetic methods, by reducing them to the Thue equation or to its generalisations over relative fields. However the bounds obtained in this way are not quite satisfactory in the general case, and we again turn to exponential equations to obtain better results. We give an analysis of the hyperelliptic equation and of integer and S-integer points on elliptic curves.

This thesis details the concepts behind several bilinear pairings, its implementations and importante. Unlike more tradicional presentations, we focus on the more general case, using the Picard Group defined over hyperelliptic curves, not focusing on particular cases, such as elliptic curves, genus 2 curves, or particular classes of curves.

A generalized bc-system associated to a Hermitian vector bundle over a Riemann surface is introduced in close analogy to the usual rank one case. Some of the geometric analogies to the well-known case are studied. In particular, if there are no zero-modes, the “non-Abelian” theta divisor appears. In the general case where zero-modes exist, it seems to be more difficult to find a natural description. It is discussed in detail why no such system exists in low genus that is on the Riemann sphere and on elliptic curves.

We introduce Four $$\mathbb {Q}$$ , a high-security, high-performance elliptic curve that targets the 128-bit security level. At the highest arithmetic level, cryptographic scalar multiplications on Four $$\mathbb {Q}$$ can use a four-dimensional Gallant-Lambert-Vanstone decomposition to minimize the total number of elliptic curve group operations. At the group arithmetic level, Four $$\mathbb {Q}$$ admits the use of extended twisted Edwards coordinates and can therefore exploit the fastest known elliptic curve addition formulas over large prime characteristic fields. Finally, at the finite field level, arithmetic is performed modulo the extremely fast Mersenne prime $$p=2^{127}-1$$ . We show that this powerful combination facilitates scalar multiplications that are significantly faster than all prior works. On Intel’s Haswell, Ivy Bridge and Sandy Bridge architectures, our software computes a variable-base scalar multiplication in 59,000, 71,000 cycles and 74,000 cycles, respectively; and, on the same platforms, our software computes a Diffie-Hellman shared secret in 92,000, 110,000 cycles and 116,000 cycles, respectively.

In this paper, we present Microblaze-based parallel architectures of Elliptic Curve Scalar Multiplication (ECSM) computation for embedded Elliptic Curve Cryptosystem (ECC) on Xilinx FPGA. The proposed implementations support arbitrary Elliptic Curve (EC) forms defined over large prime field ([Formula: see text]) with different security-level sizes. ECSM is performed using Montgomery Power Ladder (MPL) algorithm in Chudnovsky projective coordinates system. At the low abstraction level, Montgomery Modular Multiplication (MMM) is considered as the critical operation. It is implemented within a hardware Accelerator MMM (AccMMM) core based on the modified high radix, [Formula: see text] MMM algorithm. The efficiency of our parallel implementations is achieved by the combination of the mixed SW/HW approach with Multi Processor System on Programmable Chip (MPSoPC) design. The integration of multi MicroBlaze processor in single architecture allows not only the flexibility of the overall system but also the exploitation of the parallelism in ECSM computation with several degrees. The Virtex-5 parallel implementations of 256-bit and 521-bis ECSM computations run at 100[Formula: see text]MHZ frequency and consume between 2,739 and 6,533 slices, 22 and 72 RAMs and between 16 and 48 DSP48E cores. For the considered security-level sizes, the delays to perform single ECSM are between 115[Formula: see text]ms and 14.72[Formula: see text]ms.

The GLV method of Gallant, Lambert, and Vanstone (CRYPTO 2001) computes any multiple kP of a point P of prime order n lying on an elliptic curve with a low-degree endomorphism ? (called GLV curve) over $\mathbb{F}_{p}$ as $$kP = k_1P + k_2\varPhi(P) \quad\text{with } \max \bigl\{ |k_1|,|k_2| \bigr\} \leq C_1\sqrt{n} $$ for some explicit constant C 1>0. Recently, Galbraith, Lin, and Scott (EUROCRYPT 2009) extended this method to all curves over $\mathbb{F}_{p^{2}}$ which are twists of curves defined over $\mathbb{F}_{p}$ . We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over $\mathbb{F}_{p^{2}}$ , a four-dimensional decomposition together with fast endomorphisms ?,? over $\mathbb{F}_{p^{2}}$ acting on the group generated by a point P of prime order n, resulting in a proven decomposition for any scalar k?[1,n] given by $$kP=k_1P+ k_2\varPhi(P)+ k_3\varPsi(P) + k_4\varPsi\varPhi(P) \quad \text{with } \max_i \bigl(|k_i| \bigr) 0. Remarkably, taking the best C 1,C 2, we obtain C 2/C 1<412, independently of the curve, ensuring in theory an almost constant relative speedup. In practice, our experiments reveal that the use of the merged GLV---GLS approach supports a scalar multiplication that runs up to 1.5 times faster than the original GLV method. We then improve this performance even further by exploiting the Twisted Edwards model and show that curves originally slower may become extremely efficient on this model. In addition, we analyze the performance of the method on a multicore setting and describe how to efficiently protect GLV-based scalar multiplication against several side-channel attacks. Our implementations improve the state-of-the-art performance of scalar multiplication on elliptic curves over large prime characteristic fields for a variety of scenarios including side-channel protected and unprotected cases with sequential and multicore execution.

1. We set out in this paper to examine and classify all general types of quadric transformation of one space of four dimensions S 4 into another space of four dimensions S 4 '; and we shall obtain also, with certain restrictions, a method for finding quickly the reverse transformation in any given case. We shall show that this method is immediately applicable to all the most general cases, and we shall use it to obtain the reverse transformations in all such cases and in several important particular cases as well. In the course of the work many new varieties of primals are introduced in a natural way; and, in particular, a good deal of information is obtained about systems of cubic and quartic primals through certain well known surfaces—by transforming such systems to much simpler ones. For example, all cubic primals in [4] through a projected Veronese surface are transformed to cubic cones having a fixed vertex and passing through a rational normal quartic curve; and it appears further that the cubic primals in question are all 10-nodal or Segre cubics. I should like to express here my sincere thanks to Prof. H. F. Baker for his help and encouragement in the preparation of this paper.

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.