SIDH with masked torsion point images

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

M-SIDH and MD-SIDH: countering SIDH attacks by masking information

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\u27s 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 (and in particular, not to SIKE). This security picture dramatically changed in August 2022 with new attacks by Castryck-Decru, Maino-Martindale and Robert. Like prior attacks on unbalanced versions, these new attacks exploit torsion point information provided in the SIDH protocol. Crucially however, the new attacks embed the isogeny problem into a similar isogeny problem in a higher dimension to also affect the balanced parameters. As a result of these works, the SIKE algorithm is now fully broken both in theory and in practice. Given the considerable interest attracted by SIKE and related protocols in recent years, it is natural to seek countermeasures to the new attacks. In this paper, we introduce two such countermeasures based on partially hiding the isogeny degrees and torsion point information in the SIDH protocol. We present a preliminary analysis of the resulting schemes including non-trivial generalizations of prior attacks. Based on this analysis we suggest parameters for our M-SIDH variant with public key sizes of 4434, 7037 and 9750 bytes respectively for NIST security levels 1, 3, 5

SCALLOP:Scaling the CSI-FiSh

International audienceWe 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 by — arbitrary group elements, which is a requirement in, e.g., the CSI-FiSh signature scheme by Beullens, Kleinjung and Vercauteren. The index-calculus algorithm used in CSI-FiSh to compute the class-group structure has complexity L(1/2), ruling out class groups much larger than CSIDH-512, a limitation that is particularly problematic in light of the ongoing debate regarding the quantum security of cryptographic group actions.Hoping to solve this issue, we consider the class group of a quadratic order of large prime conductor inside an imaginary quadratic field of small discriminant. This family of quadratic orders lets us easily determine the size of the class group, and, by carefully choosing the conductor, even exercise significant control on it—in particular supporting highly smooth choices. Although evaluating the resulting group action still has subexponential asymptotic complexity, a careful choice of parameters leads to a practical speedup that we demonstrate in practice for a security level equivalent to CSIDH-1024, a parameter currently firmly out of reach of index-calculus-based methods. However, our implementation takes 35 seconds (resp. 12.5 minutes) for a single group-action evaluation at a CSIDH-512-equivalent (resp. CSIDH-1024-equivalent) security level, showing that, while feasible, the SCALLOP group action does not achieve realistically usable performance yet

SCALLOP: scaling the CSI-FiSh

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 by— arbitrary group elements, which is a requirement in, e.g., the CSI-FiSh signature scheme by Beullens, Kleinjung and Vercauteren. The index-calculus algorithm used in CSI-FiSh to compute the class-group structure has complexity L(1/2), ruling out class groups much larger than CSIDH-512, a limitation that is particularly problematic in light of the ongoing debate regarding the quantum security of cryptographic group actions. Hoping to solve this issue, we consider the class group of a quadratic order of large prime conductor inside an imaginary quadratic field of small discriminant. This family of quadratic orders lets us easily determine the size of the class group, and, by carefully choosing the conductor, even exercise significant control on it— in particular supporting highly smooth choices. Although evaluating the resulting group action still has subexponential asymptotic complexity, a careful choice of parameters leads to a practical speedup that we demonstrate in practice for a security level equivalent to CSIDH-1024, a parameter currently firmly out of reach of index-calculus-based methods. However, our implementation takes 35 seconds (resp. 12.5 minutes) for a single group-action evaluation at a CSIDH-512-equivalent (resp. CSIDH-1024-equivalent) security level, showing that, while feasible, the SCALLOP group action does not achieve realistically usable performance yet

Failing to hash into supersingular isogeny graphs

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 $\ell$-isogeny graph which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd's of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces; and (v) using quantum random walks.Comment: 33 pages, 7 figure

A New Adaptive Attack on SIDH

The SIDH key exchange is the main building block of SIKE, the only isogeny based scheme involved in the NIST standardization process. In 2016, Galbraith et al. presented an adaptive attack on SIDH. In this attack, a malicious party manipulates the torsion points in his public key in order to recover an honest party’s static secret key, when having access to a key exchange oracle. In 2017, Petit designed a passive attack (which was improved by de Quehen et al. in 2020) that exploits the torsion point information available in SIDH public key to recover the secret isogeny when the endomorphism ring of the starting curve is known. In this paper, firstly, we generalize the torsion point attacks by de Quehen et al. Secondly, we introduce a new adaptive attack vector on SIDH-type schemes. Our attack uses the access to a key exchange oracle to recover the action of the secret isogeny on larger subgroups. This leads to an unbalanced SIDH instance for which the secret isogeny can be recovered in polynomial time using the generalized torsion point attacks. Our attack is different from the GPST adaptive attack and constitutes a new cryptanalytic tool for isogeny based cryptography. This result proves that the torsion point attacks are relevant to SIDH (Disclaimer: this result is applicable to SIDH-type schemes only, not to SIKE.) parameters in an adaptive attack setting. We suggest attack parameters for some SIDH primes and discuss some countermeasures.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

SimS: A Simplification of SiGamal

At Asiacrypt 2020, Moriya et al. introduced two new IND-CPA secure supersingular isogeny based Public Key Encryption (PKE) protocols: SiGamal and C-SiGamal. Unlike the PKEs canonically derived from SIDH and CSIDH, the new protocols provide IND-CPA security without the use of hash functions. SiGamal and C-SiGamal are however not IND-CCA secure. Moriya et al. suggested a variant of SiGamal that could be IND-CCA secure, but left its study as an open problem. In this paper, we revisit the protocols introduced by Moriya et al. First, we show that the SiGamal variant suggested by Moriya et al. for IND-CCA security is, in fact, not IND-CCA secure. Secondly, we propose a new isogeny-based PKE protocol named SimS, obtained by simplifying SiGamal. SimS has smaller public keys and ciphertexts than (C-)SiGamal and it is more efficient. We prove that SimS is IND-CCA secure under CSIDH security assumptions and one Knowledge of Exponent-type assumption we introduce. Interestingly, SimS is also much closer to the CSIDH protocol, facilitating a comparison between SiGamal and CSIDH.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

SHealS and HealS: Isogeny-Based PKEs from a Key Validation Method for SIDH

In 2016, Galbraith et al. presented an adaptive attack on the SIDH key exchange protocol. In SIKE, one applies a variant of the Fujisaki-Okamoto transform to force Bob to reveal his encryption key to Alice, which Alice then uses to re-encrypt Bob’s ciphertext and verify its validity. Therefore, Bob can not reuse his encryption keys. There have been two other proposed countermeasures enabling static-static private keys: k-SIDH and its variant by Jao and Urbanik. These countermeasures are relatively expensive since they consist in running multiple parallel instances of SIDH. In this paper, firstly, we propose a new countermeasure to the GPST adaptive attack on SIDH. Our countermeasure does not require key disclosure as in SIKE, nor multiple parallel instances as in k-SIDH. We translate our countermeasure into a key validation method for SIDH-type schmes. Secondly, we use our key validation to design HealSIDH, an efficient SIDH-type static-static key interactive exchange protocol. Thirdly, we derive a PKE scheme SHealS using HealSIDH. SHealS uses larger primes compared to SIKE, has larger keys and ciphertexts, but only 4 isogenies are computed in a full execution of the scheme, as opposed to 5 isogenies in SIKE. We prove that SHealS is IND-CPA secure relying on a new assumption we introduce and we conjecture its IND-CCA security. We suggest HealS, a variant of SHealS using a smaller prime, providing smaller keys and ciphertexts. As a result, HealSIDH is a practically efficient SIDH based (interactive) key exchange incorporating a “direct” countermeasure to the GPST adaptive attack.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

M-SIDH and MD-SIDH:countering SIDH attacks by masking information

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 (and in particular, not to SIKE). This security picture dramatically changed in August 2022 with new attacks by Castryck-Decru, Maino-Martindale and Robert. Like prior attacks on unbalanced versions, these new attacks exploit torsion point information provided in the SIDH protocol. Crucially however, the new attacks embed the isogeny problem into a similar isogeny problem in a higher dimension to also affect the balanced parameters. As a result of these works, the SIKE algorithm is now fully broken both in theory and in practice. Given the considerable interest attracted by SIKE and related protocols in recent years, it is natural to seek countermeasures to the new attacks. In this paper, we introduce two such countermeasures based on partially hiding the isogeny degrees and torsion point information in the SIDH protocol. We present a preliminary analysis of the resulting schemes including non-trivial generalizations of prior attacks. Based on this analysis we suggest parameters for our M-SIDH variant with public key sizes of 4434, 7037 and 9750 bytes respectively for NIST security levels 1, 3, 5.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

SCALLOP: scaling the CSI-FiSh

International audienceWe 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 by — arbitrary group elements, which is a requirement in, e.g., the CSI-FiSh signature scheme by Beullens, Kleinjung and Vercauteren. The index-calculus algorithm used in CSI-FiSh to compute the class-group structure has complexity L(1/2), ruling out class groups much larger than CSIDH-512, a limitation that is particularly problematic in light of the ongoing debate regarding the quantum security of cryptographic group actions.Hoping to solve this issue, we consider the class group of a quadratic order of large prime conductor inside an imaginary quadratic field of small discriminant. This family of quadratic orders lets us easily determine the size of the class group, and, by carefully choosing the conductor, even exercise significant control on it—in particular supporting highly smooth choices. Although evaluating the resulting group action still has subexponential asymptotic complexity, a careful choice of parameters leads to a practical speedup that we demonstrate in practice for a security level equivalent to CSIDH-1024, a parameter currently firmly out of reach of index-calculus-based methods. However, our implementation takes 35 seconds (resp. 12.5 minutes) for a single group-action evaluation at a CSIDH-512-equivalent (resp. CSIDH-1024-equivalent) security level, showing that, while feasible, the SCALLOP group action does not achieve realistically usable performance yet