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AbstractA tangential quadrilateral is a convex quadrilateral whose sides are simultaneously tangent to a single circle. In this paper, the primary objective is to construct rational tangential quadrilaterals characterized by having rational area, as well as rational side and diagonal lengths. We relate the existence of such tangential quadrilaterals to properties of a certain elliptic curve. Studying the curve, we are able to construct an infinite family of rational tangential quadrilaterals. [ABSTRACT FROM AUTHOR]

We extend the criterion of Kawatani and Okawa for indecomposability of the derived category of a smooth projective variety to arbitrary schemes. For relative schemes, we also give a criterion for the nonexistence of semiorthogonal decompositions that are linear over the base. These criteria are based on the base loci of the global or relative dualizing complexes. [ABSTRACT FROM AUTHOR]

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs (m, n) the locus of polarised abelian surfaces of type (1, d) that contain two complementary elliptic curve of exponents m, n, denoted E d (m , n) is non-empty. We show that if d is square-free, the locus E d (m , n) is an irreducible surface (if non-empty). We also show that the loci E d (d , d) can have many components if d is an odd square. As an application, we show that for a genus 3 curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings f i : C → E i , i = 1 , 2 , 3 satisfy lcm (deg (f 1) , deg (f 2)) = lcm (deg (f 1) , deg (f 3)) = lcm (deg (f 2) , deg (f 3)) . [ABSTRACT FROM AUTHOR]

Let N be a positive integer. Let H be a subgroup of GL (2 , Z / N Z) of level N and let E be an elliptic curve defined over the rationals with j E ≠ 0 , 1728 . Then the image ρ ¯ E , N Gal Q ¯ / Q , of the mod-N Galois representation attached to E, is conjugate to a subgroup of H if and only if E corresponds to a non-cuspidal rational point on the modular curve X H generated by H. In this article, we are interested when ρ ¯ E , N Gal Q ¯ / Q is conjugate to H. More precisely, we classify all subgroups H of GL (2 , Z / N Z) that are direct products of groups of distinct prime-power level for which X H contains infinitely many non-cuspidal rational points but there is no elliptic curve E / Q such that ρ ¯ E , N Gal Q ¯ / Q is conjugate to H itself. [ABSTRACT FROM AUTHOR]

Following the work of Lalín and Mittal on the Mahler measure over arbitrary tori, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in two variables when they do not vanish on the integration torus. We establish certain relations between the standard Mahler measure and the generalized Mahler measure of such polynomials. Later we focus our investigation on a tempered family of polynomials originally studied by Boyd, namely Q r (x , y) = x + 1 x + y + 1 y + r with r ∈ C , and apply our results to this family. For the r = 4 case, we explicitly calculate the generalized Mahler measure of Q 4 over any arbitrary torus in terms of special values of the Bloch–Wigner dilogarithm. Finally, we extend our results to the several variable setting. [ABSTRACT FROM AUTHOR]

By searching for good quadratic sections, we construct a new example of an elliptic curve over function field Q (k) with rank at least 4 containing a torsion point of order four. [ABSTRACT FROM AUTHOR]

A one-third century ago, as a means to speed up the elliptic curve method (ECM) for integer factoring, Montgomery suggested using a special elliptic curve form over prime fields and developed an addition chain to compute scalar multiplication on them, which nowadays are famous as Montgomery curves and Montgomery ladder. Kim et al. (http://eprint.iacr.org/2017/669. 2017) and Kim et al. (Adv Math Commun https://doi.org/10.3934/amc.2020090. 2020) found the Montgomery ladder very efficient on every short Weierstrass curve, leading to the most efficient regular scalar multiplication algorithms, which was further improved by Hamburg (https://ches.2017.rump.cr.yp.to/. 2020) and Hamburg (http://eprint.iacr.org/2020/437. 2020). However, the efficiency of the Montgomery ladder in general Montgomery curves remained not improved at all since firstly presented by Montgomery. This paper addresses the long-standing Elliptic Curve Cryptography (ECC) problem. The topic of this article is considered one of the topics that have attracted much attention from the cryptographic community following the launch of a multi-year project called "Post-Quantum Cryptography Standardization" by the National Institute of Standards and Technology (NIST) and also thanks partly to featuring one of the smallest keys of any algorithm known in the literature that is conjectured to be quantum resistant. To the best of our knowledge, this article provides, for the first time after Peter L. Montgomery's, an improvement of arithmetic in general Montgomery curves, including point doubling and differential addition, which are the most fundamental operations in the context of ECC and supersingular isogeny-based primitives such as Supersingular Isogeny Diffie–Hellman (SIDH) or Supersingular Isogeny Key Encapsulation (SIKE), as well as ECM. [ABSTRACT FROM AUTHOR]

Let d ≥ 1 be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8. [ABSTRACT FROM AUTHOR]

The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical question for post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a finite field. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard” quantum algorithm for HSP on finite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare” as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantum algorithm that uses mm coset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on the success probability of a quantum algorithm that uses mm coset samples to solve DCP.

We characterize the possible groups E(Z∕NZ)E\left({\mathbb{Z}}/N{\mathbb{Z}}) arising from elliptic curves over Z∕NZ{\mathbb{Z}}/N{\mathbb{Z}} in terms of the groups E(Fp)E\left({{\mathbb{F}}}_{p}), with pp varying among the prime divisors of NN. This classification is achieved by showing that the infinity part of any elliptic curves over Z∕peZ{\mathbb{Z}}/{p}^{e}{\mathbb{Z}} is a Z∕peZ{\mathbb{Z}}/{p}^{e}{\mathbb{Z}}-torsor, of which a generator is exhibited. As a first consequence, when E(Z∕NZ)E\left({\mathbb{Z}}/N{\mathbb{Z}}) is a pp-group, we provide an explicit and sharp bound on its rank. As a second consequence, when N=peN={p}^{e} is a prime power and the projected curve E(Fp)E\left({{\mathbb{F}}}_{p}) has trace one, we provide an isomorphism attack to the elliptic curve discrete logarithm problem, which works only by means of finite ring arithmetic.

Finding elliptic curves with high ranks has been the focus of much research. Recently, with the goal of generating elliptic curves with a large rank, some authors used large integers n which have many divisors, amongst which one can find divisors d such that d + n / d is a perfect square. This strategy is in itself a motivation for studying the function τ □ (n) which counts the number of divisors d of an integer n for which d + n / d is a perfect square. We show that ∑ n ≤ x τ □ (n) = c □ x 3 / 4 + O (x) for some explicit constant c □ . Moreover, letting ρ 1 (n) : = max { d ∣ n : d ≤ n } and ρ 2 (n) : = min { d ∣ n : d ≥ n } stand for the middle divisors of n, we show that the order of magnitude of the number of positive integers n ≤ x for which ρ 1 (n) + ρ 2 (n) is a perfect square is x 3 / 4 / log x . [ABSTRACT FROM AUTHOR]

We prove that several Feigin–Odesskii Poisson brackets associated with normal elliptic curves in P n are compatible if and only if they are contained in a scroll or in a Veronese surface in P 5 (with an exception of one case when n = 3 ). In the case n = 3 we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves E 1 and E 2 . [ABSTRACT FROM AUTHOR]

A formula is proved for the number of linear factors and irreducible cubic factors over F l of the Hasse invariant H ^ 7 , l (a) of the elliptic curve E 7 (a) in Tate normal form, on which the point (0, 0) has order 7, as a polynomial in the parameter a, in terms of the class number of the imaginary quadratic field K = Q (- l) . Conjectural formulas are stated for the numbers of quadratic and sextic factors of H ^ 7 , l (a) of certain specific forms in terms of the class number of Q (- 7 l) , which are shown to imply a recent conjecture of Nakaya on the number of linear factors over F l of the supersingular polynomial s s l (7 ∗) (X) corresponding to the Fricke group Γ 0 ∗ (7) . [ABSTRACT FROM AUTHOR]

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in P 5 . We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of I ( Sec (C 6)) . We study the images C p and C p q of a sextic C 6 under the projection from a general point P ∈ P 5 and a general line P Q ¯ ⊂ P 5 . In particular, we show that C p is k-normal for all k ≥ 2 and I (C p) is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that C p q is k-normal for all k ≥ 3 and I (C p q) is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce a modified k-Fibonacci-like sequence defined on an elliptic curve and prove Binet’s formula for this sequence. Moreover, we give a new encryption scheme using this sequence.

We construct an efficient dynamic group signature (or more generally an accountable ring signature) from isogeny and lattice assumptions. Our group signature is based on a simple generic construction that can be instantiated by cryptographically hard group actions such as the CSIDH group action or an MLWE-based group action. The signature is of size O (log N) , where N is the number of users in the group. Our idea builds on the recent efficient OR-proof by Beullens, Katsumata, and Pintore (Asiacrypt'20), where we efficiently add a proof of valid ciphertext to their OR-proof and further show that the resulting non-interactive zero-knowledge proof system is online extractable. Our group signatures satisfy more ideal security properties compared to previously known constructions, while simultaneously having an attractive signature size. The signature size of our isogeny-based construction is an order of magnitude smaller than all previously known post-quantum group signatures (e.g., 6.6 KB for 64 members). In comparison, our lattice-based construction has a larger signature size (e.g., either 126 KB or 89 KB for 64 members depending on the satisfied security property). However, since the O (·) -notation hides a very small constant factor, it remains small even for very large group sizes, say 2 20 . [ABSTRACT FROM AUTHOR]

It is well-known that the Parshin–Arakelov theorem implies the Mordell conjecture over complex function fields by a covering construction of Parshin. Via a similar map in the context of integral points on elliptic curves over function fields, we explain how to obtain a short geometric proof of a uniform version of Siegel's theorem. Our technique also allows us to establish a uniform quantitative result on the set-theoretic intersection of curves with the singular divisor in the compact moduli space of stable curves. [ABSTRACT FROM AUTHOR]

The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields F q , with q ≡ 3 (mod 4) , we introduce a finite field analogue AGM F q that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e., tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of F q -points, which can be used to prove that each swarm has at least (1 / 2 − ε) q jellyfish. This interpretation also gives a description of the class numbers of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish. [ABSTRACT FROM AUTHOR]

The algebras Q n , k (E , τ) introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers n > k ≥ 1 , a complex elliptic curve E, and a point τ ∈ E . The main result in this paper is that Q n , k (E , τ) has the same Hilbert series as the polynomial ring on n variables when τ is not a torsion point. We also show that Q n , k (E , τ) is a Koszul algebra, hence of global dimension n when τ is not a torsion point, and, for all but countably many τ , Q n , k (E , τ) is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining Q n , k (E , τ) is the image of an operator R τ (τ) that belongs to a family of operators R τ (z) : C n ⊗ C n → C n ⊗ C n , z ∈ C , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter. [ABSTRACT FROM AUTHOR]

This article makes an important contribution to solving the long-standing problem of whether all elliptic curves can be equipped with a hash function (indifferentiable from a random oracle) whose running time amounts to one exponentiation in the basic finite field Fq{{\mathbb{F}}}_{q}. More precisely, we construct a new indifferentiable hash function to any ordinary elliptic Fq{{\mathbb{F}}}_{q}-curve Ea{E}_{a} of j-invariant 1728 with the cost of extracting one quartic root in Fq{{\mathbb{F}}}_{q}. As is known, the latter operation is equivalent to one exponentiation in finite fields with which we deal in practice. In comparison, the previous fastest random oracles to Ea{E}_{a} require to perform two exponentiations in Fq{{\mathbb{F}}}_{q}. Since it is highly unlikely that there is a hash function to an elliptic curve without any exponentiations at all (even if it is supersingular), the new result seems to be unimprovable.

In this paper, our aim is to explicitly calculate the relations amongst the rational equivalence classes of Brill–Noether loci and describe the algebra generated by the same in the Chow ring of the moduli space of semistable bundles on elliptic curve. In general, the relations obtained are dependent on the degree of the embeddings involved. We also provide an example of a particular fixed determinant moduli space where the relations obtained are embedding independent. [ABSTRACT FROM AUTHOR]

In this paper we derive formulas for the scalar multiplication by n map, denoted [n], on the Hessian model of elliptic curve. This enables to characterize n-torsion points on this curve. The computation involves three families of polynomials P n , Q n and V n and we show some properties on the coefficients and degrees of these polynomials. We also show some functional equations satisfied by these polynomials. As application we provide a type of mean-value theorem for the Hessian elliptic curve. [ABSTRACT FROM AUTHOR]

In this work, we use division polynomials on Edwards elliptic curves to construct elliptic nets of rank one called elliptic divisibility sequence on this curve. Introduced by Morgan Ward, elliptic divisibility sequence on elliptic curve are used for solving the elliptic curve discrete logarithm problem (ECDLP). Moved on rank 2, Elliptic Nets algorithm via elliptic nets associated to Weierstrass elliptic curves are useful for the computation of bilinear maps (called pairings) defined on the groups of points of elliptic curves and very useful to the construction of cryptographic protocols. [ABSTRACT FROM AUTHOR]

We investigate the solvability of a class of Diophantine equations y 2 + z 2 + B = A x 3 + x y z , defined over the set of integers Z where A and B are positive integers with certain conditions. As a consequence, we obtain a family of elliptic curves whose torsion subgroup of Mordell–Weil group over Q is trivial. [ABSTRACT FROM AUTHOR]

Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].

Let x3 + ax + b be an irreducible cubic polynomial over p. Then by using transformations x → x + j and x → j–1x, where j ∈ F*p, we can generate new irreducible cubic polynomials. In this paper, we use the elliptic curve equation y2 = x3 + ax + b and some transformations to generate an irreducible cubic polynomial, say Q. We generate the public and private keys in the elliptic curve digital signature algorithm (ECDSA), by defining the isomorphism between.Finally, we discuss the security of keys and the validity of signature. [ABSTRACT FROM AUTHOR]

In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves E 1 × E 2 without complex multiplication are integers. By studying elliptic curves on E 1 × E 2 we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on E 1 × E 2 are integers if and only if the minimal degree of an isogeny E 1 → E 2 equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on E 1 × E 2 . [ABSTRACT FROM AUTHOR]

The main attack against static-key supersingular isogeny Diffie–Hellman (SIDH) is the Galbraith–Petit–Shani–Ti (GPST) attack, which also prevents the application of SIDH to other constructions such as non-interactive key-exchange. In this paper, we identify and study a specific assumption on which the GPST attack relies that does not necessarily hold in all circumstances. We show that in some circumstances the attack fails to recover part of the secret key. We also characterize the conditions necessary for the attack to fail and show that it rarely happens in real cases. We give a link with collisions in the Charles-Goren-Lauter (CGL) hash function.

Fog computing is a computing structure which is distributed in nature. Low latency, reasonably low communication overhead and ability to support real time applications are the reasons for which fog computing approach said to provide better performance than cloud computing. Although, it is an extension of the cloud computing. Fog computing also inherits some critical security and privacy issues of cloud computing. Secure key management and user authentication are among the key issues faced by fog computing. Various schemes with probable solutions of these issues have been proposed by many authors in this context. Among them, a notable scheme has been presented by Wajid et al. known as SAKA-FC, where authors used three-factor authentication with privacy preservation for remote user based on ECC, hash functions, fuzzy extractor and symmetric bivariate polynomial function. This paper analyses the SAKA-FC protocol and found that it is not resilient against fog server insider attack, message intercept attack and replay attack. Consequently, an improved, lightweight and secure authentication scheme in context of fog-centric IoT communication is proposed in this paper to eradicate all the above mentioned security shortfalls of Wajid et al scheme. The proposed scheme is verified using mathematical security analysis and simulated using AVISPA which proves that the proposed scheme prevents all pertinent security threats. The performance analysis of our scheme proves its effectiveness over other related existing schemes in this context. [ABSTRACT FROM AUTHOR]

The widespread adoption of cloud computing enables the end-users to leverage convenient sharing, unlimited storage and on-demand access to big data. The extensive combination of servers, networks, users and resources necessitate secure mutual authentication protocol to verify the legitimacy of users for cloud services. Recently, Sahoo et al. and Chen et al. proposed multi-factor mutual authentication and key agreement (MAKA) protocols. However, we identify that Sahoo et al.'s protocol is prone to user linkability, replay and denial-of-service (DoS) attacks. Also, Chen et al.'s protocol is vulnerable to user linkability and known session-specific temporary information (KSSTI) attack. To mitigate these vulnerabilities, we propose a novel elliptic curve cryptography (ECC) based provably secure and privacy-preserving multi-factor authentication protocol for cloud environment. Our protocol delivers user anonymity, unlinkability, perfect forward secrecy, session key security as security and privacy authentication features. The security of our protocol is proved theoretically under Real-Or-Random (ROR) model. We validate the correctness properties of our protocol under Scyther security verification tool. The informal security analysis illustrates that our protocol resists various security attacks such as replay, DoS, KSSTI, user impersonation, server spoofing, password-guessing and privileged insider. Finally, we compare our protocol with Sahoo et al., Chen et al. and other existing relevant protocols regarding security features, communication, and computation overheads. The results illustrate that our protocol exhibits high security with reasonable communication and computational overheads than other existing relevant protocols. [ABSTRACT FROM AUTHOR]

Loops and cycles play an important role in computing endomorphism rings of supersingular elliptic curves and related cryptosystems. For a supersingular elliptic curve E defined over 𝔽p2, if an imaginary quadratic order O can be embedded in End(E) and a prime L splits into two principal ideals in O, we construct loops or cycles in the supersingular L-isogeny graph at the vertices which are next to j(E) in the supersingular ℓ-isogeny graph where ℓ is a prime different from L. Next, we discuss the lengths of these cycles especially for j(E) = 1728 and 0. Finally, we also determine an upper bound on primes p for which there are unexpected 2-cycles if ℓ doesn’t split in O.

Elliptic curves are typically defined by Weierstrass equations. Given a kernel, the well-known Vélu's formula shows how to explicitly write down an isogeny between Weierstrass curves. However, it is not clear how to do the same on other forms of elliptic curves without isomorphisms mapping to and from the Weierstrass form. Previous papers have shown some isogeny formulas for (twisted) Edwards, Huff, and Montgomery forms of elliptic curves. Continuing this line of work, this paper derives explicit formulas for isogenies between elliptic curves in (twisted) Hessian form. In addition, we examine the numbers of operations in the base field to compute the formulas. In comparison with other isogeny formulas, we note that our formulas for twisted Hessian curves have the lowest costs for processing the kernel and our X-affine formula has the lowest cost for processing an input point in affine coordinates.

We introduce in this paper the algorithmic aspect of elliptic curves together with their applications. We also recall one of the promising application in the field of zero knowledge proofs with concrete implementations. [ABSTRACT FROM AUTHOR]

Let F q be a finite field and E b : y 2 = x 3 + b be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that b ∈ F q and q ≢ 1 (mod 27) . For example, these conditions are fulfilled for the curve BLS12-381 ( b = 4 ). It is a de facto standard in the real-world pairing-based cryptography at the moment. This article provides a new constant-time hash function H : { 0 , 1 } ∗ → E b (F q) indifferentiable from a random oracle. Its main advantage is the fact that H computes only one exponentiation in F q . In comparison, the previous fastest constant-time indifferentiable hash functions to E b (F q) compute two exponentiations in F q . In particular, applying H to the widely used BLS multi-signature with m different messages, the verifier should perform only m exponentiations rather than 2m ones during the hashing phase. [ABSTRACT FROM AUTHOR]

We introduce a category of 𝓞-oriented supersingular elliptic curves and derive properties of the associated oriented and nonoriented ℓ-isogeny supersingular isogeny graphs. As an application we introduce an oriented supersingular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.

A positive integer N is called a θ -congruent number if there is a θ -triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N r 2 - s 2 , where θ ∈ (0 , π) , cos (θ) = s / r , and 0 ≤ | s | < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a θ -congruent number if and only if the elliptic curve E N θ : y 2 = x (x + (r + s) N) (x - (r - s) N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N ≠ 1 , 2 , 3 , 6 is a θ -congruent number if and only if rank of E N θ (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ -triangle for a θ -congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E N θ (Q) . Then, based on the addition of two distinct points in E N θ (Q) , we provide a way to find new rational θ -triangles for the θ -congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E N θ (Q) with corresponding rational θ -triangles. [ABSTRACT FROM AUTHOR]

In a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).