Any isogeny between two supersingular elliptic curves can be defined over $\mathbb{F}_{p^2}$, however, this does not imply that computing such isogenies can be done with field operations in $\mathbb{F}_{p^2}$. In fact, the kernel generators of such isogenies are defined over extension fields of $\mathbb{F}_{p^2}$, generically with extension degree linear to the isogeny degree. Most algorithms related to isogeny computations are only efficient when the extension degree is small. This leads to...

The Supersingular Isogeny Diffie-Hellman (SIDH) scheme is a public key cryptosystem that was submitted to the National Institute of Standards and Technology's competition for the standardization of post-quantum cryptography protocols. The private key in SIDH consists of an isogeny whose degree is a prime power. In July 2022, Castryck and Decru discovered an attack that completely breaks the scheme by recovering Bob's secret key, using isogenies between higher dimensional abelian varieties to...

In 2020, Castryck-Decru-Smith constructed a hash function, using the (2,2)-isogeny graph of superspecial principally polarized abelian surfaces. In their construction, the initial surface was chosen from vertices very "close" to the square of a supersingular elliptic curve with a known endomorphism ring. In this paper, we introduce an algorithm for detecting a collision on their hash function. Under some heuristic assumptions, the time complexity and space complexity of our algorithm are...

A crucial ingredient for many cryptographic primitives such as key exchange protocols and advanced signature schemes is a commutative group action where the structure of the underlying group can be computed efficiently. SCALLOP provides such a group action, based on oriented supersingular elliptic curves. We present PEARL-SCALLOP, a variant of SCALLOP that changes several parameter and design choices, thereby improving on both efficiency and security and enabling feasible parameter...

Proving knowledge of a secret isogeny has recently been proposed as a means to generate supersingular elliptic curves of unknown endomorphism ring, but is equally important for cryptographic protocol design as well as for real world deployments. Recently, Cong, Lai and Levin (ACNS'23) have investigated the use of general-purpose (non-interactive) zero-knowledge proof systems for proving the knowledge of an isogeny of degree $2^k$ between supersingular elliptic curves. In particular, their...

We construct lollipops of pairing-friendly elliptic curves, which combine pairing-friendly chains with pairing-friendly cycles. The cycles inside these lollipops allow for unbounded levels of recursive pairing-based proof system composition, while the chains leading into these cycles alleviate a significant drawback of using cycles on their own: the only known cycles of pairing-friendly elliptic curves force the initial part of the circuit to be arithmetised on suboptimal (much larger)...

In this paper we study several computational problems related to current post-quantum cryptosystems based on isogenies between supersingular elliptic curves. In particular we prove that the supersingular isogeny path and endomorphism ring problems are unconditionally equivalent under polynomial time reductions. We show that access to a factoring oracle is sufficient to solve the Quaternion path problem of KLPT and prove that these problems are equivalent, where previous results either...

We extend the usual ideal action on oriented elliptic curves to a (Hermitian) module action on oriented (polarised) abelian varieties. Oriented abelian varieties are naturally enriched in $R$-modules, and our module action comes from the canonical power object construction on categories enriched in a closed symmetric monoidal category. In particular our action is canonical and gives a fully fledged symmetric monoidal action. Furthermore, we give algorithms to compute this action in practice,...

Non-Interactive Timed Commitment schemes (NITC) allow to open any commitment after a specified delay $t_{\mathrm{fd}}$. This is useful for sealed bid auctions and as primitive for more complex protocols. We present the first NITC without repeated squaring or theoretical black box algorithms like NIZK proofs or one-way functions. It has fast verification, almost arbitrary delay and satisfies IND-CCA hiding and perfect binding. Our protocol is based on isogenies between supersingular elliptic...

The security of certain post-quantum isogeny-based cryptographic schemes relies on the ability to provably and efficiently compute isogenies between supersingular elliptic curves without leaking information about the isogeny other than its domain and codomain. Earlier work in this direction give mathematical proofs of knowledge for the isogeny, and as a result when computing a chain of $n$ isogenies each proceeding node must verify the correctness of the proof of each preceding node, which...

Dimension 4 isogenies have first been introduced in cryptography for the cryptanalysis of Supersingular Isogeny Diffie-Hellman (SIDH) and have been used constructively in several schemes, including SQIsignHD, a derivative of SQIsign isogeny based signature scheme. Unlike in dimensions 2 and 3, we can no longer rely on the Jacobian model and its derivatives to compute isogenies. In dimension 4 (and higher), we can only use theta-models. Previous works by Romain Cosset, David Lubicz and Damien...

This paper proposes a fast and efficient FPGA-based hardware-software co-design for the supersingular isogeny key encapsulation (SIKE) protocol controlled by a custom RISC-V processor. Firstly, we highly optimize the core unit, the polynomial-based field arithmetic logic unit (FALU), with the proposed fast convolution-like multiplier (FCM) to significantly reduce the resource consumption while still maintaining low latency and constant time for all the four SIKE parameters. Secondly, we pack...

We introduce a new tool for the study of isogeny-based cryptography, namely pairings which are sesquilinear (conjugate linear) with respect to the $\mathcal{O}$-module structure of an elliptic curve with CM by an imaginary quadratic order $\mathcal{O}$. We use these pairings to study the security of problems based on the class group action on collections of oriented ordinary or supersingular elliptic curves. This extends work of of both (Castryck, Houben, Merz, Mula, Buuren, Vercauteren,...

One of the most promising avenues for realizing scalable proof systems relies on the existence of 2-cycles of pairing-friendly elliptic curves. Such a cycle consists of two elliptic curves E/GF(p) and E'/GF(q) that both have a low embedding degree and also satisfy q = #E and p = #E'. These constraints turn out to be rather restrictive; in the decade that has passed since 2-cycles were first proposed for use in proof systems, no new constructions of 2-cycles have been found. In this paper,...

The Deuring correspondence is a correspondence between supersingular elliptic curves and quaternion orders. Under this correspondence, an isogeny between elliptic curves corresponds to a quaternion ideal. This correspondence plays an important role in isogeny-based cryptography and several algorithms to compute an isogeny corresponding to a quaternion ideal (ideal-to-isogeny algorithms) have been proposed. In particular, SQIsign is a signature scheme based on the Deuring correspondence and...

In isogeny-based cryptography, bilinear pairings are regarded as a powerful tool in various applications, including key compression, public-key validation and torsion basis generation. However, in most isogeny-based protocols, the performance of pairing computations is unsatisfactory due to the high computational cost of the Miller function. Reducing the computational expense of the Miller function is crucial for enhancing the overall performance of pairing computations in isogeny-based...

Adaptor signatures can be viewed as a generalized form of the standard digital signature schemes where a secret randomness is hidden within a signature. Adaptor signatures are a recent cryptographic primitive and are becoming an important tool for blockchain applications such as cryptocurrencies to reduce on-chain costs, improve fungibility, and contribute to off-chain forms of payment in payment-channel networks, payment-channel hubs, and atomic swaps. However, currently used adaptor...

In this note we propose a variant (with four sub-variants) of the Charles--Goren--Lauter (CGL) hash function using Lattès maps over finite fields. These maps define dynamical systems on the projective line. The underlying idea is that these maps ``hide'' the $j$-invariants in each step in the isogeny chain, similar to the Merkle--Damgård construction. This might circumvent the problem concerning the knowledge of the starting (or ending) curve's endomorphism ring, which is known to create...

In general the discrete logarithm problem is a hard problem in the elliptic curve cryptography, and the best known solving algorithm have exponential running time. But there exists a class of curves, i.e. supersingular elliptic curves, whose discrete logarithm problem has a subexponential solving algorithm called the MOV attack. In 1999, the cost of the MOV reduction is still computationally expensive due to the power of computers. We analysis the cost of the MOV reduction and the discrete...

In 2021, Sterner proposed a commitment scheme based on supersingular isogenies. For this scheme to be binding, one relies on a trusted party to generate a starting supersingular elliptic curve of unknown endomorphism ring. In fact, the knowledge of the endomorphism ring allows one to compute an endomorphism of degree a power of a given small prime. Such an endomorphism can then be split into two to obtain two different messages with the same commitment. This is the reason why one needs a...

Recent work by Arpin, Chen, Lauter, Scheidler, Stange, and Tran counted the number of cycles of length $r$ in supersingular $\ell$-isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine. We provide formulas for both the number of such cycles, and the average number as $p \to \infty$, with $\ell$ and $r$ fixed. In particular, we show that when $r$ is not a power of $2$, cycles of length $r$ are disproportionately likely to occur along the...

We present a new post-quantum Public Key Encryption scheme (PKE) named Supersingular Isogeny Lollipop Based Encryption or SILBE. SILBE is obtained by leveraging the generalised lollipop attack of Castryck and Vercauteren on the M-SIDH Key exchange by Fouotsa, Moriya and Petit. Doing so, we can in fact make SILBE a post-quantum secure Updatable Public Key Encryption scheme (UPKE). SILBE is in fact the first isogeny-based UPKE which is not based on group actions. Hence, SILBE overcomes the...

At EUROCRYPT’23, Castryck and Decru, Maino et al., and Robert present efficient attacks against supersingular isogeny Diffie-Hellman key exchange protocol (SIDH). Drawing inspiration from these attacks, Andrea Basso, Luciano Maino, and Giacomo Pope introduce FESTA, an isogeny-based trapdoor function, along with a corresponding IND-CCA secure public key encryption (PKE) protocol at ASIACRYPT’23. FESTA incorporates either a diagonal or circulant matrix into the secret key to mask torsion...

This work introduces several algorithms related to the computation of orientations in endomorphism rings of supersingular elliptic curves. This problem boils down to representing integers by ternary quadratic forms, and it is at the heart of several results regarding the security of oriented-curves in isogeny-based cryptography. Our main contribution is to show that there exists efficient algorithms that can solve this problem for quadratic orders of discriminant $n$ up to $O(p^{4/3})$....

The polynomial attacks on SIDH by Castryck, Decru, Maino, Martindale and Robert have shown that, while the general isogeny problem is still considered unfeasible to break, it is possible to efficiently compute a secret isogeny when given its degree and image on enough torsion points. A natural response from many researchers has been to propose SIDH variants where one or both of these possible extra pieces of information is masked in order to obtain schemes for which a polynomial attack is...

Finding isogenies between supersingular elliptic curves is a natural algorithmic problem which is known to be equivalent to computing the curves' endomorphism rings. When the isogeny is additionally required to have a specific known degree $d$, the problem appears to be somewhat different in nature, yet its hardness is also required in isogeny-based cryptography. Let $E_1,E_2$ be supersingular elliptic curves over $\mathbb{F}_{p^2}$. We present improved classical and quantum...

Isogeny computations in CSIDH (Asiacrypt 2018) are described using a commutative group G acting on the set of supersingular elliptic curves. The commutativity property gives CSIDH enough flexibility to allow the creation of many cryptographic primitives and protocols. Nevertheless, these operations are limited and more complex applications have not yet been proposed. When calling the composition of two group elements of G addition, our goal in this work is to explore exponentiation,...

We present a verifiable delay function based on isogenies of supersingular elliptic curves, using Deuring correspondence and computation of endomorphism rings for the delay. For each input x a verifiable delay function has a unique output y and takes a predefined time to evaluate, even with parallel computing. Additionally, it generates a proof by which the output can efficiently be verified. In our approach the input is a path in the 2-isogeny graph and the output is the maximal order...

Isogeny-based cryptography is one of the candidates for post-quantum cryptography. One of the benefits of using isogeny-based cryptography is its compactness. In particular, a key exchange scheme SIDH allowed us to use a $4\lambda$-bit prime for the security parameter $\lambda$. Unfortunately, SIDH was broken in 2022 by some studies. After that, some isogeny-based key exchange and public key encryption schemes have been proposed; however, most of these schemes use primes whose sizes are...

In 2023, Basso, Maino, and Pope proposed FESTA (Fast Encryption from Supersingular Torsion Attacks), an isogeny-based public-key encryption (PKE) protocol that uses the SIDH attack for decryption. In the same paper, they proposed a parameter for that protocol, but the parameter requires high-degree isogeny computations. In this paper, we introduce QFESTA (Quaternion Fast Encapsulation from Supersingular Torsion Attacks), a new variant of FESTA that works with better parameters using...

The identity-based signature, initially introduced by Shamir [Sha84], plays a fundamental role in the domain of identity-based cryptography. It offers the capability to generate a signature on a message, allowing any user to verify the authenticity of the signature using the signer's identifier information (e.g., an email address), instead of relying on a public key stored in a digital certificate. Another significant concept in practical applications is the threshold signature, which serves...

Given a supersingular elliptic curve $E$ and a non-scalar endomorphism $\alpha$ of $E$, we prove that the endomorphism ring of $E$ can be computed in classical time about $\text{disc}(\mathbb{Z}[\alpha])^{1/4}$ , and in quantum subexponential time, assuming the generalised Riemann hypothesis. Previous results either had higher complexities, or relied on heuristic assumptions. Along the way, we prove that the Primitivisation problem can be solved in polynomial time (a problem previously...

We define and analyze the Commutative Isogeny Hidden Number Problem which is the natural analogue of the Hidden Number Problem in the CSIDH and CSURF setting. In short, the task is as follows: Given two supersingular elliptic curves \(E_A\), \(E_B\) and access to an oracle that outputs some of the most significant bits of the \({\mathsf{CDH}}\) of two curves, an adversary must compute the shared curve \(E_{AB}={\mathsf{CDH}}(E_A,E_B)\). We show that we can recover \(E_{AB}\) in...

The supersingular Endomorphism Ring problem is the following: given a supersingular elliptic curve, compute all of its endomorphisms. The presumed hardness of this problem is foundational for isogeny-based cryptography. The One Endomorphism problem only asks to find a single non-scalar endomorphism. We prove that these two problems are equivalent, under probabilistic polynomial time reductions. We prove a number of consequences. First, assuming the hardness of the endomorphism ring...

Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $\mathfrak{O}$-orientable for a fixed imaginary quadratic order $\mathfrak{O}$ provides non-trivial information towards computing an endomorphism corresponding to the $\mathfrak{O}$-orientation....

In this paper, we introduce $\mathsf{DeuringVUF}$, a new Verifiable Unpredictable Function (VUF) protocol based on isogenies between supersingular curves. The most interesting application of this VUF is $\mathsf{DeuringVRF}$ a post-quantum Verifiable Random Function (VRF). The main advantage of this new scheme is its compactness, with combined public key and proof size of roughly 450 bytes, which is orders of magnitude smaller than other generic purpose post-quantum VRF...

A number of supersingular isogeny based cryptographic protocols require the endomorphism ring of the initial elliptic curve to be either unknown or random in order to be secure. To instantiate these protocols, Basso et al. recently proposed a secure multiparty protocol that generates supersingular elliptic curves defined over $\mathbb{F}_{p^2}$ of unknown endomorphism ring as long as at least one party acts honestly. However, there are many protocols that specifically require curves defined...

Pairings are useful tools in isogeny-based cryptography and have been used in SIDH/SIKE and other protocols. As a general technique, pairings can be used to move problems about points on curves to elements in finite fields. However, until now, their applicability was limited to curves over fields with primes of a specific shape and pairings seemed too costly for the type of primes that are nowadays often used in isogeny-based cryptography. We remove this roadblock by optimizing pairings for...

The Supersingular Isogeny Diffie-Hellman (SIDH) protocol has been the main and most efficient isogeny-based encryption protocol, until a series of breakthroughs led to a polynomial-time key-recovery attack. While some countermeasures have been proposed, the resulting schemes are significantly slower and larger than the original SIDH. In this work, we propose a new countermeasure technique that leads to significantly more efficient and compact protocols. To do so, we introduce the...

The Isogeny to Endomorphism Ring Problem (IsERP) asks to compute the endomorphism ring of the codomain of an isogeny between supersingular curves in characteristic $p$ given only a representation for this isogeny, i.e. some data and an algorithm to evaluate this isogeny on any torsion point. This problem plays a central role in isogeny-based cryptography; it underlies the security of pSIDH protocol (ASIACRYPT 2022) and it is at the heart of the recent attacks that broke the SIDH key...

We introduce FESTA, an efficient isogeny-based public-key encryption (PKE) protocol based on a constructive application of the SIDH attacks. At its core, FESTA is based on a novel trapdoor function, which uses an improved version of the techniques proposed in the SIDH attacks to develop a trapdoor mechanism. Using standard transformations, we construct an efficient PKE that is IND-CCA secure in the QROM. Additionally, using a different transformation, we obtain the first isogeny-based PKE...

We present an attack on SIDH utilising isogenies between polarized products of two supersingular elliptic curves. In the case of arbitrary starting curve, our attack (discovered independently from [CD22]) has subexponential complexity, thus significantly reducing the security of SIDH and SIKE. When the endomorphism ring of the starting curve is known, our attack (here derived from [CD22]) has polynomial-time complexity assuming the generalised Riemann hypothesis. Our attack applies to any...

We present a tightly secure identity-based signature (IBS) scheme based on the supersingular isogeny problems. Although Shaw and Dutta proposed an isogeny-based IBS scheme with provable security, the security reduction is non-tight. For an IBS scheme with concrete security, the tightness of its security reduction affects the key size and signature size. Hence, it is reasonable to focus on a tight security proof for an isogeny-based IBS scheme. In this paper, we propose an isogeny-based IBS...

Time-based cryptographic primitives such as Time-Lock Puzzles (TLPs) and Verifiable Delay Functions (VDFs) have proven to be pivotal in several areas of cryptography. All existing candidate constructions, however, guarantee time-delays based on the average hardness of sequential computational problems. This means that any algorithmic or hardware improvement affects parameter choices and may turn deployed systems insecure. To address this issue, we investigate how to build time-based...

Constructing a supersingular elliptic curve whose endomorphism ring is isomorphic to a given quaternion maximal order (one direction of the Deuring correspondence) is known to be polynomial-time assuming the generalized Riemann hypothesis [KLPT14; Wes21], but notoriously daunting in practice when not working over carefully selected base fields. In this work, we speed up the computation of the Deuring correspondence in general characteristic, i.e., without assuming any special form of the...

We propose a novel methodology to obtain $B$lind signatures that is fundamentally based on the idea of hiding part of the underlying plain signatures under a $Z$ero-knowledge argument of knowledge of the whole signature (hence the shorthand, $BZ$). Our proposal is necessarily non-black-box and stated in the random oracle model. We illustrate the technique by describing two instantiations: a classical setting based on the traditional discrete logarithm assumption, and a post-quantum setting...

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on ordinary curves, due to the supposed inefficiency of the supersingular case. While this was true a decade ago, the recent advances in the study of supersingular curves through the Deuring correspondence motivated by isogeny-based cryptography has...

We present SCALLOP: SCALable isogeny action based on Oriented supersingular curves with Prime conductor, a new group action based on isogenies of supersingular curves. Similarly to CSIDH and OSIDH, we use the group action of an imaginary quadratic order’s class group on the set of oriented supersingular curves. Compared to CSIDH, the main benefit of our construction is that it is easy to compute the class-group structure; this data is required to uniquely represent — and efficiently act...

Generating supersingular elliptic curves of unknown endomorphism ring has been a problem vexing isogeny-based cryptographers for several years. A recent development has proposed a trusted setup protocol to generate such a curve, where each participant generates and proves knowledge of an isogeny. Thus, the construction of efficient proofs of knowledge of isogeny has developed new interest. Historically, the isogeny community has assumed that obtaining isogeny proofs of knowledge from...

The SIDH protocol is an isogeny-based key exchange protocol using supersingular isogenies, designed by Jao and De Feo in 2011. The protocol underlies the SIKE algorithm which advanced to the fourth round of NIST's post-quantum standardization project in May 2022. The algorithm was considered very promising: indeed the most significant attacks against SIDH were meet-in-the-middle variants with exponential complexity, and torsion point attacks which only applied to unbalanced parameters...

Generating a supersingular elliptic curve such that nobody knows its endomorphism ring is a notoriously hard task, despite several isogeny-based protocols relying on such an object. A trusted setup is often proposed as a workaround, but several aspects remain unclear. In this work, we develop the tools necessary to practically run such a distributed trusted-setup ceremony. Our key contribution is the first statistically zero-knowledge proof of isogeny knowledge that is compatible with any...

The security guarantees of most isogeny-based protocols rely on the computational hardness of finding an isogeny between two supersingular isogenous curves defined over a prime field $\mathbb{F}_q$ with $q$ a power of a large prime $p$. In most scenarios, the isogeny is known to be of degree $\ell^e$ for some small prime $\ell$. We call this problem the Supersingular Fixed-Degree Isogeny Path (SIPFD) problem. It is believed that the most general version of SIPFD is not solvable faster than...

Binary elliptic curves are elliptic curves defined over finite fields of characteristic 2. On software platforms that offer carryless multiplication opcodes (e.g. pclmul on x86), they have very good performance. However, they suffer from some drawbacks, in particular that non-supersingular binary curves have an even order, and that most known formulas for point operations have exceptional cases that are detrimental to safe implementation. In this paper, we show how to make a prime order...

We address three main open problems concerning the use of radical isogenies, as presented by Castryck, Decru and Vercauteren at Asiacrypt 2020, in the computation of long chains of isogenies of fixed, small degree between elliptic curves over finite fields. Firstly, we present an interpolation method for finding radical isogeny formulae in a given degree $N$, which by-passes the need for factoring division polynomials over large function fields. Using this method, we are able to push the...

In the context of quantum-resistant cryptography, cryptographic group actions offer an abstraction of isogeny-based cryptography in the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) setting. In this work, we revisit the security of two previously proposed natural protocols: the Group Action Hashed ElGamal key encapsulation mechanism (GA-HEG KEM) and the Group Action Hashed Diffie-Hellman non-interactive key-exchange (GA-HDH NIKE) protocol. The latter protocol has already been...

Although the supersingular isogeny Diffie-Hellman (SIDH) protocol is one of the most promising post-quantum cryptosystems, it is significantly slower than its main counterparts due to the underlying large smooth-degree isogeny computation. In this work, we address the problem of evaluating and constructing a strategy for computing the large smooth-degree isogeny in the multi-processor setting by formulating them as scheduling problems with dependencies. The contribution of this work is...

We propose a countermeasure to the Castryck-Decru attack on SIDH. The attack heavily relies on the images of torsion points. The main input to our countermeasure consists in masking the torsion point images in SIDH in a way they are not exploitable in the attack, but can be used to complete the key exchange. This comes with a change in the form the field characteristic and a considerable increase in the parameter sizes.

We present an attack on SIDH which does not require any endomorphism information on the starting curve. Our attack has subexponential complexity thus significantly reducing the security of SIDH and SIKE; our analysis and preliminary implementation suggests that our algorithm will be feasible for the Microsoft challenge parameters $p = 2^{110}3^{67}-1$ on a regular computer. Our attack applies to any isogeny-based cryptosystem that publishes the images of points under the secret isogeny, for...

Elliptic curves are abelian varieties of dimension one; the two-dimensional analogue are abelian surfaces. In this work we present an algorithm to compute $(2^n,2^n)$-isogenies of abelian surfaces defined over finite fields. These isogenies are the natural generalization of $2^n$-isogenies of elliptic curves. Our algorithm is designed to be used in higher-dimensional variants of isogeny-based cryptographic protocols such as G2SIDH which is a genus-$2$ version of the Supersingular Isogeny...

We present an efficient key recovery attack on the Supersingular Isogeny Diffie-Hellman protocol (SIDH). The attack is based on Kani's "reducibility criterion" for isogenies from products of elliptic curves and strongly relies on the torsion point images that Alice and Bob exchange during the protocol. If we assume knowledge of the endomorphism ring of the starting curve then the classical running time is polynomial in the input size (heuristically), apart from the factorization of a small...

SQISign is an isogeny-based signature scheme that has short keys and signatures and is expected to be a post-quantum scheme. Its security depends on the hardness of the problem to find an isogeny between given two elliptic curves over $\mathbb{F}_{p^2}$, where $p$ is a large prime. For efficiency reasons, a public key in SQISign is taken from a set of supersingular elliptic curves with a particular property. In this paper, we investigate the security related to public keys in SQISign. First,...

Many public-key cryptographic protocols, notably non-interactive key exchange (NIKE), require incoming public keys to be validated to mitigate some adaptive attacks. In CSIDH, an isogeny-based post-quantum NIKE, a key is deemed legitimate if the given Montgomery coefficient specifies a supersingular elliptic curve over the prime field. In this work, we survey the current supersingularity tests used for CSIDH key validation, and implement and measure two new alternative algorithms. Our...

SIDH is a key exchange algorithm proposed by Jao and De Feo that is conjectured to be post-quantum secure. The majority of work based on an SIDH framework uses elliptic curves in Montgomery form; this includes the original work by Jao, De Feo and Plût and the sate of the art implementation of SIKE. Elliptic curves in twisted Edwards form have also been used due to their efficient elliptic curve arithmetic, and complete Edwards curves have been used for their benefit of providing added...

We propose a new Diffie-Hellman-like Non-Interactive Key Exchange that uses the Lattice Isomorphisms as a building block. Our proposal also relies on a group action structure, implying a similar security setup as in the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) protocol where Kuperberg's algorithm applies. We short label our scheme as LIKE. As with the original Diffie-Hellman protocol, our proposed scheme is also passively secure. We provide a proof-of-concept constant-time...

Isogeny-based cryptography is a promising approach for post-quantum cryptography. The best-known protocol following that approach is the supersingular isogeny Diffie-Hellman protocol (SIDH); this protocol was turned into the CCA-secure key encapsulation mechanism SIKE, which was submitted to and remains in the third round of NIST's post-quantum standardization process as an ``alternate'' candidate. Isogeny-based cryptography generally relies on the conjectured hardness of computing an...

We consider the use of supersingular abelian surfaces in cryptography. Several generalisations of well-known cryptographic schemes and constructions based on supersingular elliptic curves to the 2-dimensional setting of superspecial abelian surfaces have been proposed. The computational assumptions in the superspecial 2-dimensional case can be reduced to the corresponding 1-dimensional problems via a product decomposition by observing that every superspecial abelian surface is non-simple and...

The paper concerns several theoretical aspects of oriented supersingular $\ell$-isogeny volcanoes and their relationship to closed walks in the supersingular $\ell$-isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular $\ell$-isogeny volcanoes over $\overline{\mathbb{F}}_p$ (up to conjugation of the orientations), and isogeny cycles (non-backtracking closed walks which are not powers of smaller walks) of the supersingular $\ell$-isogeny...

We consider the problem of sampling random supersingular elliptic curves over finite fields of cryptographic size (SRS problem). The currently best-known method combines the reduction of a suitable complex multiplication (CM) $j$-invariant and a random walk over some supersingular isogeny graph. Unfortunately, this method is not suitable for numerous cryptographic applications because it gives information about the endomorphism ring of the generated curve. This motivates a stricter version...

An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular ℓ-isogeny graph which does not reveal the endomorphism ring, or a path to a curve of known...

We propose a novel approach that generalizes interleaved modular multiplication algorithms for the computation of sums of products over large prime fields. This operation has widespread use and is at the core of many cryptographic applications. The method reformulates the widely used lazy reduction technique, crucially avoiding the need for storage and computation of "double-precision" operations. Moreover, it can be easily adapted to the different methods that exist to compute modular...

In this article, we prove a generic lower bound on the number of $\mathfrak{O}$-orientable supersingular curves over $\mathbb{F}_{p^2}$, i.e curves that admit an embedding of the quadratic order $\mathfrak{O}$ inside their endomorphism ring. Prior to this work, the only known effective lower-bound is restricted to small discriminants. Our main result targets the case of fundamental discriminants and we derive a generic bound using the expansion properties of the supersingular isogeny graphs....

The Deuring correspondence defines a bijection between isogenies of supersingular elliptic curves and ideals of maximal orders in a quaternion algebra. We present a new algorithm to translate ideals of prime-power norm to their corresponding isogenies --- a central task of the effective Deuring correspondence. The new method improves upon the algorithm introduced in 2021 by De Feo, Kohel, Leroux, Petit and Wesolowski as a building-block of the SQISign signature scheme. SQISign is the...

In this paper, we generalise the SIDH fault attack and the SIDH loop-abort fault attacks on supersingular isogeny cryptosystems (genus-1) to genus-2. Genus-2 isogeny-based cryptosystems are generalisations of its genus-1 counterpart, as such, attacks on the latter are believed to generalise to the former. The point perturbation attack on supersingular elliptic curve isogeny cryptography has been shown to be practical. We show in this paper that this fault attack continues to be practical...

In supersingular isogeny-based cryptography, the path-finding problem reduces to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? It is known that a small endomorphism enables polynomial-time path-finding and endomorphism ring computation (Love-Boneh [36]). An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this paper, we use the volcano structure of the oriented supersingular isogeny graph to take...

The Oriented Supersingular Isogeny Diffie-Hellman is a post-quantum key exchange scheme recently introduced by Colò and Kohel. It is based on the group action of an ideal class group of a quadratic imaginary order on a subset of supersingular elliptic curves, and in this sense it can be viewed as a generalization of the popular isogeny based key exchange CSIDH. From an algorithmic standpoint, however, OSIDH is quite different from CSIDH. In a sense, OSIDH uses class groups which are more...

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 $\mathbb{F}_{\!q}$. More precisely, we construct a new indifferentiable hash function to any ordinary elliptic $\mathbb{F}_{\!q}$-curve $E_a$ of $j$-invariant $1728$ with the cost of extracting one quartic root in...

This paper focuses on isogeny representations, defined as ways to evaluate isogenies and verify membership to the language of isogenous supersingular curves (the set of triples $D,E_1,E_2$ with a cyclic isogeny of degree $D$ between $E_1$ and $E_2$). The tasks of evaluating and verifying isogenies are fundamental for isogeny-based cryptography. Our main contribution is the design of the suborder representation, a new isogeny representation targeted at the case of (big) prime...

We study two important families of problems in isogeny-based cryptography and how they relate to each other: computing the endomorphism ring of supersingular elliptic curves, and inverting the action of class groups on oriented supersingular curves. We prove that these two families of problems are closely related through polynomial-time reductions, assuming the generalised Riemann hypothesis. We identify two classes of essentially equivalent problems. The first class corresponds to the...

Currently, public-key compression of supersingular isogeny Diffie-Hellman (SIDH) and its variant, supersingular isogeny key encapsulation (SIKE) involve pairing computation and discrete logarithm computation. Both of them require large storage for precomputation to accelerate the performance. In this paper, we propose a novel method to compute only three discrete logarithms instead of four, in exchange for computing a lookup table efficiently. We also suggest another alternative method to...

In 2016, the National Institute of Standards and Technology (NIST) initiated a standardization process among the post-quantum secure algorithms. Forming part of the alternate group of candidates after Round 2 of the process is the Supersingular Isogeny Key Encapsulation (SIKE) mechanism which attracts with the smallest key sizes offering post-quantum security in scenarios of limited bandwidth and memory resources. Even further reduction of the exchanged information is offered by the...

We give a new algorithm for finding an isogeny from a given supersingular elliptic curve $E/\mathbb{F}_{p^2}$ to a subfield elliptic curve $E'/\mathbb{F}_p$, which is the bottleneck step of the Delfs-Galbraith algorithm for the general supersingular isogeny problem. Our core ingredient is a novel method of rapidly determining whether a polynomial $f \in L[X]$ has any roots in a subfield $K \subset L$, while crucially avoiding expensive root-finding algorithms. In the special case when...

This work focuses on concrete cryptanalysis of the isogeny-based cryptosystems SIDH/SIKE under realistic memory/storage constraints. More precisely, we are solving the problem of finding an isogeny of a given smooth degree between two given supersingular elliptic curves. Recent works by Adj et al. (SAC 2018), Costello et al. (PKC 2020), Longa et al. (CRYPTO 2021) suggest that parallel "memoryless" golden collision search by van Oorschot-Wiener (JoC 1999) is the best realistic approach for...

Supersingular isogeny Diffe-Hellman (SIDH) is attractive for its relatively small public key size, but it is still unsatisfactory due to its effciency, compared to other post-quantum proposals. In this paper, we focus on the performance of SIDH when the starting curve is $E_6 : y^2 = x^3 + 6x^2 + x$, which is fixed in Round-3 SIKE implementation. Inspired by the previous work, we present several tricks to accelerate key generation of SIDH and each process of SIKE. Our experimental results...

A ring signature is a digital signature scheme that allows identifying a group of possible signers without revealing the identity of the actual signer. In this paper, we first present a post-quantum sigma protocol for a ring that relies on the supersingular isogeny-based interactive zero-knowledge identification scheme proposed by De Feo, Jao, and Plût in 2014. Then, we construct a ring signature from the proposed sigma protocol for a ring by applying the Fiat-Shamir transform. In order to...

In this paper, we investigate the problem of constructing postquantum-secure verifiable delay functions (VDFs), particularly based on supersingular isogenies. Isogeny-based VDF constructions have been proposed before, but since verification relies on pairings, they are broken by quantum computers. We propose an entirely different approach using succinct non-interactive arguments (SNARGs), but specifically tailored to the arithmetic structure of the isogeny setting to achieve good asymptotic...

In the effort to transition cryptographic primitives and protocols to quantum-resistant alternatives, an interesting and useful challenge is found in the Signal protocol. The initial key agreement component of this protocol, called X3DH, has so far proved more subtle to replace - in part due to the unclear security model and properties the original protocol is designed for. This paper defines a formal security model for the original signal protocol, in the context of the standard eCK and CK+...

The digital signature schemes that have been proposed so far in the setting of the Supersingular Isogeny Diffie-Hellman scheme (SIDH) were obtained by applying the Fiat-Shamir transform - and a quantum-resistant analog, the Unruh transform - to an interactive identification protocol introduced by De Feo, Jao and Plût. The security of the resulting schemes is therefore deduced from that of the base identification protocol. In this paper, we revisit the proofs that have appeared in the...

In this work we present two commitment schemes based on hardness assumptions arising from supersingular elliptic curve isogeny graphs, which possess strong security properties. The first is based on the CGL hash function while the second is based on the SIDH framework, both of which require a trusted third party for the setup phrase. The proofs of security of these protocols depend on properties of non-backtracking random walks on regular graphs. The optimal efficiency of these protocols...

We show that the soundness proof for the De Feo-Jao-Plut identification scheme (the basis for supersingular isogeny Diffie--Hellman (SIDH) signatures) contains an invalid assumption, and we provide a counterexample for this assumption---thus showing the proof of soundness is invalid. As this proof was repeated in a number of works by various authors, multiple pieces of literature are affected by this result. Due to the importance of being able to prove knowledge of an SIDH key (for example,...

In recent years, the isogeny-based protocol, namely supersingular isogeny Diffe-Hellman (SIDH) has become highly attractive for its small public key size. In addition, public-key compression makes supersingular isogeny key encapsulation scheme (SIKE) more competitive in the NIST post-quantum cryptography standardization effort. However, compared to other post-quantum protocols, the computational cost of SIDH is relatively high, and so is public-key compression. On the other hand, the storage...

We present a polynomial-time adaptive attack on the genus-2 variant of the SIDH protocol (G2SIDH) and describe an improvement to its secret selection procedure. G2SIDH is a generalisation of the Supersingular Isogeny Diffie--Hellman key exchange into the genus-2 setting which was proposed by Flynn and Ti. G2SIDH is able to achieve the same security as SIDH while using fields a third of the size. We analyze the keyspace of G2SIDH and achieve an improvement to the secret selection by using...

We investigate the isogeny graphs of supersingular elliptic curves over \(\mathbb{F}_{p^2}\) equipped with a \(d\)-isogeny to their Galois conjugate. These curves are interesting because they are, in a sense, a generalization of curves defined over \(\mathbb{F}_p\), and there is an action of the ideal class group of \(\mathbb{Q}(\sqrt{-dp})\) on the isogeny graphs. We investigate constructive and destructive aspects of these graphs in isogeny-based cryptography, including generalizations of...

We prove that the path-finding problem in $\ell$-isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis. The presumed hardness of these problems is foundational for isogeny-based cryptography. As an essential tool, we develop a rigorous algorithm for the quaternion analog of the path-finding problem, building upon the heuristic method of Kohel, Lauter, Petit...

The threat of quantum computers has sparked the development of a new kind of cryptography to resist their attacks. Isogenies between elliptic curves are one of the tools used for such cryptosystems. They are championed by SIKE (Supersingular isogeny key encapsulation), an "alternate candidate" of the third round of the NIST Post-Quantum Cryptography Standardization Process. While all candidates are believed to be mathematically secure, their implementations may be vulnerable to hardware...

In this paper, we introduce a new method to prove the knowledge of an isogeny of given degree between two supersingular elliptic curves. Our approach can be extended to verify the evaluation of the secret isogeny on some points of the domain. The main advantage of this new proof of knowledge is its compactness which is orders of magnitude better than existing proofs of isogeny knowledge. The principle of our method is to reveal some well-chosen endomorphisms and does not constitute a...

We cryptanalyse the SIDH-based oblivious pseudorandom function from supersingular isogenies proposed at Asiacrypt'20 by Boneh, Kogan and Woo. To this end, we give an attack on an assumption, the auxiliary one-more assumption, that was introduced by Boneh et al. and we show that this leads to an attack on the oblivious PRF itself. The attack breaks the pseudorandomness as it allows adversaries to evaluate the OPRF without further interactions with the server after some initial OPRF...

Software implementations of cryptographic algorithms are slow but highly flexible and relatively easy to implement. On the other hand, hardware implementations are usually faster but provide little flexibility and require a lot of time to implement efficiently. In this paper, we develop a hybrid software-hardware implementation of the third round of Supersingular Isogeny Key Encapsulation (SIKE), a post-quantum cryptography algorithm candidate for NIST. We implement an isogeny field...

To mark the 10-year anniversary of supersingular isogeny Diffie-Hellman, I will touch on 10 points in defense and support of the SIKE protocol, including the rise of classical hardness, the fact that quantum computers do not seem to offer much help in solving the underlying problem, and the importance of concrete cryptanalytic clarity. In the final section I present the two SIKE challenges: $55k USD is up for grabs for the solutions of mini instances that, according to the SIKE team's...

Although isogeny-based cryptographic schemes enjoy the lowest key sizes amongst current post-quantum cryptographic candidates, they unfortunately come at a high computational cost, which makes their deployment on the ever-growing number of resource-constrained devices difficult. Speeding up the expensive post-quantum cryptographic operations by delegating these computations from a weaker client to untrusted powerful external servers is a promising approach. Following this, we present in this...

Key-oblivious encryption (KOE) is a newly developed cryptographic primitive that randomizes the public keys of an encryption scheme in an oblivious manner. It has applications in designing accountable tracing signature (ATS) that facilitates the group manager to revoke the anonymity of traceable users in a group signature while preserving the anonymity of non-traceable users. Despite of its importance and strong application, KOE has not received much attention in the literature. In this...