Volume 15, Issue 1

Secret sharing is an important building block in cryptography. All explicit secret sharing schemes which are known to have optimal complexity are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give a partial answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual.

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond( V n ) of the Vandermonde matrix V n associated to the n th cyclotomic polynomial. We prove some results on the singular values of V n and, in particular, we determine Cond( V n ) for n = 2 k p ℓ , where k , ℓ ≥ 0 are integers and p is an odd prime number.

In this paper, we show a “direct” equivalence between certain authentication codes and robust threshold schemes. It was previously known that authentication codes and robust threshold schemes are closely related to similar types of designs, but direct equivalences had not been considered in the literature. Our new equivalences motivate the consideration of a certain “key-substitution attack.” We study this attack and analyze it in the setting of “dual authentication codes.” We also show how this viewpoint provides a nice way to prove properties and generalizations of some known constructions.

Let Ω be a finite set of finitary operation symbols. We initiate the study of (weakly) pseudo-free families of computational Ω -algebras in arbitrary varieties of Ω -algebras. A family ( H d | d ∈ D ) of computational Ω -algebras (where D ⊆ {0, 1} * ) is called polynomially bounded (resp., having exponential size) if there exists a polynomial η such that for all d ∈ D , the length of any representation of every h ∈ H d is at most η(|d|)( resp., |Hd|≤2η(|d|)). $\eta (|d|)\left( \text{ resp}\text{., }\left| {{H}_{d}} \right|\le {{2}^{\eta (|d|)}} \right).$ First, we prove the following trichotomy: (i) if Ω consists of nullary operation symbols only, then there exists a polynomially bounded pseudo-free family; (ii) if Ω = Ω 0 ∪ { ω }, where Ω 0 consists of nullary operation symbols and the arity of ω is 1, then there exist an exponential-size pseudo-free family and a polynomially bounded weakly pseudo-free family; (iii) in all other cases, the existence of polynomially bounded weakly pseudo-free families implies the existence of collision-resistant families of hash functions. In this trichotomy, (weak) pseudo-freeness is meant in the variety of all Ω -algebras. Second, assuming the existence of collision-resistant families of hash functions, we construct a polynomially bounded weakly pseudo-free family and an exponential-size pseudo-free family in the variety of all m -ary groupoids, where m is an arbitrary positive integer.

Lattice sieving in two dimensions has proven to be an indispensable practical aid in integer factorization and discrete log computations involving the number field sieve. The main contribution of this article is to show that a different method of lattice sieving in three dimensions will provide a significant speedup in medium characteristic. Our method is to use the successive minima and shortest vectors of the lattice instead of transition vectors to iterate through lattice points. We showcase the new method by a record computation in a 133-bit subgroup of Fp6 ${{\mathbb{F}}_{{{p}^{6}}}}$, with p 6 having 423 bits. Our overall timing is nearly 3 times faster than the previous record of a 132-bit subgroup in a 422-bit field. The approach generalizes to dimensions 4 or more, overcoming one key obstruction to the implementation of the tower number field sieve.

We analyze security properties of a two-party key-agreement protocol recently proposed by I. Anshel, D. Atkins, D. Goldfeld, and P. Gunnels, called Kayawood protocol. At the core of the protocol is an action (called E-multiplication ) of a braid group on some finite set. The protocol assigns a secret element of a braid group to each party (private key). To disguise those elements, the protocol uses a so-called cloaking method that multiplies private keys on the left and on the right by specially designed elements (stabilizers for E-multiplication). We present a heuristic algorithm that allows a passive eavesdropper to recover Alice’s private key by removing cloaking elements. Our attack has 100% success rate on randomly generated instances of the protocol for the originally proposed parameter values and for recent proposals that suggest to insert many cloaking elements at random positions of the private key. Implementation of the attack is available on GitHub.

At ProvSec 2013, Minematsu presented the circulant hash, an almost-xor universal hash using only the xor and rotation operations. The circulant hash is a variant of Carter and Wegman’s H 3 hash as well as Krawczyk’s Toeplitz hash, both of which are hashes based on matrix-vector multiplication over 𝔽 2 . In this paper we revisit the circulant hash and reinterpret it as a multiplication in the polynomial ring 𝔽 2 [ x ]/( x n + 1). This leads to simpler proofs, faster implementations in modern computer chips, and newer variants with practical implementation advantages.

S-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for ( n + 1)-bit S-boxes in [12]. We obtain the nonlinearity of ( n + 1)-bit S-boxes, and one necessary and sufficient conditions of ( n + 1)-bit S-boxes satisfying m -order resilient. Meanwhile, we also give one characterization of ( n + 1)-bit S-boxes satisfying t -order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an n -bit S-box S 0 and the ( n + 1)-bit S-box S (which is constructed by S 0 ).

ElGamal cryptosystem has emerged as one of the most important construction in Public Key Cryptography (PKC) since Diffie-Hellman key exchange protocol was proposed. However, public key schemes which are based on number theoretic problems such as discrete logarithm problem (DLP) are at risk because of the evolution of quantum computers. As a result, other non-number theoretic alternatives are a dire need of entire cryptographic community. In 2016, Saba Inam and Rashid Ali proposed a ElGamal-like cryptosystem based on matrices over group rings in ‘Neural Computing & Applications’. Using linear algebra approach, Jia et al. provided a cryptanalysis for the cryptosystem in 2019 and claimed that their attack could recover all the equivalent keys. However, this is not the case and we have improved their cryptanalysis approach and derived all equivalent key pairs that can be used to totally break the ElGamal-like cryptosystem proposed by Saba and Rashid. Using the decomposition of matrices over group rings to larger size matrices over rings, we have made the cryptanalysing algorithm more practical and efficient. We have also proved that the ElGamal cryptosystem proposed by Saba and Rashid does not achieve the security of IND-CPA and IND-CCA.

We consider a key-exchange protocol based on matrices over a tropical semiring which was recently proposed in [2]. We show that a particular private parameter of that protocol can be recovered with a simple binary search, rendering it insecure.

We generalize a protocol by Yu for comparing two integers with relatively small difference in a secure multiparty computation setting. Yu's protocol is based on the Legendre symbol. A prime number p is found for which the Legendre symbol (· | p ) agrees with the sign function for integers in a certain range {− N , . . . , N } ⊂ ℤ. This can then be computed efficiently. We generalize this idea to higher residue symbols in cyclotomic rings ℤ[ ζ r ] for r a small odd prime. We present a way to determine a prime number p such that the r -th residue symbol (· | p ) r agrees with a desired function f:A→{ζr0,…,ζrr−1} f:A \to \left\{ {\zeta _r^0, \ldots ,\zeta _r^{r - 1}} \right\} on a given small subset A ⊂ ℤ[ ζ r ], when this is possible. We also explain how to efficiently compute the r -th residue symbol in a secret shared setting.

There are many cryptographic applications of Boolean functions. Recently, research has been done on monomial rotation symmetric (MRS) functions which have useful cryptographic properties. In this paper we use the inclusion/exclusion principle to construct formulas for the weights of two subclasses of MRS functions: degree d short MRS functions and d -functions. From these results we classify bent and balanced functions of these forms.

Permutation-based modes have been established for lightweight authenticated encryption, as can be seen from the high interest in the ongoing NIST lightweight competition. However, their security is upper bounded by O ( σ 2 /2 c ) bits, where σ are the number of calls and c is the hidden capacity of the state. The development of more schemes that provide higher security bounds led to the CHES’18 proposal Beetle that raised the bound to O ( rσ /2 c ), where r is the public rate of the state. While authenticated encryption can be performed in an on-line manner, authenticated decryption assumes that the resulting plaintext is buffered and never released if the corresponding tag is incorrect. Since lightweight devices may lack the resources for buffering, additional robustness guarantees, such as integrity under release of unverified plaintexts (I nt -RUP), are desirable. In this stronger setting, the security of the established schemes, including Beetle, is limited by O ( q p q d /2 c ), where q d is the maximal number of decryption queries, and q p that of off-line primitive queries, which motivates novel approaches. This work proposes Oribatida, a permutation-based AE scheme that derives s -bit masks from previous permutation outputs to mask ciphertext blocks. Oribatida can provide a security bound of O ( rσ 2 / c + s ), which allows smaller permutations for the same level of security. It provides a security level dominated by O(σd2/2c) O(\sigma_d^2{/2^c}) under I nt -RUP adversaries, which eliminates the dependency on primitive queries. We prove its security under nonce-respecting and I nt -RUP adversaries. We show that our I nt -RUP bound is tight and show general attacks on previous constructions.

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 generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.

Extra-reductions occurring in Montgomery multiplications disclose side-channel information which can be exploited even in stringent contexts. In this article, we derive stochastic attacks to defeat Rivest-Shamir-Adleman (RSA) with Montgomery ladder regular exponentiation coupled with base blinding. Namely, we leverage on precharacterized multivariate probability mass functions of extra-reductions between pairs of (multiplication, square) in one iteration of the RSA algorithm and that of the next one(s) to build a maximum likelihood distinguisher. The efficiency of our attack (in terms of required traces) is more than double compared to the state-of-the-art. In addition to this result, we also apply our method to the case of regular exponentiation, base blinding, and modulus blinding. Quite surprisingly, modulus blinding does not make our attack impossible, and so even for large sizes of the modulus randomizing element. At the cost of larger sample sizes our attacks tolerate noisy measurements. Fortunately, effective countermeasures exist.

Boolean functions have important applications in molecular regulatory networks, engineering, cryptography, information technology, and computer science. Symmetric Boolean functions have received a lot of attention in several decades. Sensitivity and block sensitivity are important complexity measures of Boolean functions. In this paper, we study the sensitivity of elementary symmetric Boolean functions and obtain many explicit formulas. We also obtain a formula for the block sensitivity of symmetric Boolean functions and discuss its applications in elementary symmetric Boolean functions.

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 𝔽 p 2 , 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 .

Attribute-based proxy re-encryption (ABPRE), which combines the notions of proxy re-encryption (PRE) and attribute-based encryption (ABE), allows a semi-trusted proxy with re-encryption key to transform a ciphertext under a particular access policy into a ciphertext under another access policy, without revealing any information about the underlying plaintext. This primitive is very useful in applications where encrypted data need to be stored in untrusted environments, such as cloud storage. In many practical applications, and in order to address scenarios where users misbehave or the re-encryption keys are compromised, an efficient revocation mechanism is necessary for ABPRE. Previously, revocation mechanism was considered in the settings of identity-based encryption (IBE), ABE, predicate encryption (PE), and broadcast PRE, but not ABPRE, which is what we set to do in this paper. We first formalize the concept of revocable ABPRE and its security model. Then, we propose a lattice-based instantiation of revocable ABPRE. Our scheme not only supports an efficient revocation mechanism but also supports polynomial-depth policy circuits and has short private keys, where the size of the keys is dependent only on the depth of the supported policy circuits. In addition, we prove that our scheme is selectively chosen-plaintext attack (CPA) secure in the standard model, based on the learning with errors assumption.

We propose a heuristic algorithm to solve the underlying hard problem of the CSIDH cryptosystem (and other isogeny-based cryptosystems using elliptic curves with endomorphism ring isomorphic to an imaginary quadratic order 𝒪). Let Δ = Disc(𝒪) (in CSIDH, Δ = −4 p for p the security parameter). Let 0 < α < 1/2, our algorithm requires: A classical circuit of size 2O˜log(|Δ|)1−α. $2^{\tilde{O}\left(\log(|\Delta|)^{1-\alpha}\right)}.$ A quantum circuit of size 2O˜log(|Δ|)α. $2^{\tilde{O}\left(\log(|\Delta|)^{\alpha}\right)}.$ Polynomial classical and quantum memory. Essentially, we propose to reduce the size of the quantum circuit below the state-of-the-art complexity 2O˜log(|Δ|)1/2 $2^{\tilde{O}\left(\log(|\Delta|)^{1/2}\right)}$ at the cost of increasing the classical circuit-size required. The required classical circuit remains subexponential, which is a superpolynomial improvement over the classical state-of-the-art exponential solutions to these problems. Our method requires polynomial memory, both classical and quantum.

Password authenticated key establishment (PAKE) is a cryptographic primitive that allows two parties who share a low-entropy secret (a password) to securely establish cryptographic keys in the absence of public key infrastructure. We propose the first quantum-resistant password-authenticated key exchange scheme based on supersingular elliptic curve isogenies. The scheme is built upon supersingular isogeny Diffie-Hellman [15], and uses the password to generate permutations which obscure the auxiliary points. We include elements of a security proof, and discuss roadblocks to obtaining a proof in the BPR model [1]. We also include some performance results.

Recently, supersingular isogeny cryptosystems have received attention as a candidate of post-quantum cryptography (PQC). Their security relies on the hardness of solving isogeny problems over supersingular elliptic curves. The meet-in-the-middle approach seems the most practical to solve isogeny problems with classical computers. In this paper, we propose two algebraic approaches for isogeny problems of prime power degrees. Our strategy is to reduce isogeny problems to a system of algebraic equations, and to solve it by Gröbner basis computation. The first one uses modular polynomials, and the second one uses kernel polynomials of isogenies. We report running times for solving isogeny problems of 3-power degrees on supersingular elliptic curves over 𝔽 p 2 with 503-bit prime p , extracted from the NIST PQC candidate SIKE. Our experiments show that our first approach is faster than the meet-in-the-middle approach for isogeny degrees up to 3 10 .

A statistical framework applicable to Ring-LWE was outlined by Murphy and Player (IACR eprint 2019/452). Its applicability was demonstrated with an analysis of the decryption failure probability for degree-1 and degree-2 ciphertexts in the homomorphic encryption scheme of Lyubashevsky, Peikert and Regev (IACR eprint 2013/293). In this paper, we clarify and extend results presented by Murphy and Player. Firstly, we make precise the approximation of the discretisation of a Normal random variable as a Normal random variable, as used in the encryption process of Lyubashevsky, Peikert and Regev. Secondly, we show how to extend the analysis given by Murphy and Player to degree- k ciphertexts, by precisely characterising the distribution of the noise in these ciphertexts.

We revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis–Laarhoven–De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than 2 0.076 d + o ( d ) $2^{0.076d + o(d)}$ memory, and we show how to obtain time–memory trade-offs even when using less than 2 0.048 d + o ( d ) $2^{0.048d + o(d)}$ memory. We also settle the open problem of obtaining a continuous trade-off between the size of the advice and the query time complexity, as the time complexity with subexponential advice in our approach scales as d d / 2 + o ( d ) $d^{d/2 + o(d)}$ matching worst-case enumeration bounds, and achieving the same asymptotic scaling as average-case enumeration algorithms for the closest vector problem.

We initiate the study of partial key exposure in Ring-LWE (RLWE)-based cryptosystems. Specifically, we (1) Introduce the search and decision Leaky R-LWE assumptions (Leaky R-SLWE, Leaky R-DLWE), to formalize the hardness of search/decision RLWE under leakage of some fraction of coordinates of the NTT transform of the RLWE secret. (2) Present and implement an efficient key exposure attack that, given certain 1/4-fraction of the coordinates of the NTT transform of the RLWE secret, along with samples from the RLWE distribution, recovers the full RLWE secret for standard parameter settings. (3) Present a search-to-decision reduction for Leaky R-LWE for certain types of key exposure. (4) Propose applications to the security analysis of RLWE-based cryptosystems under partial key exposure.

The leftover hash lemma (LHL) is used in the analysis of various lattice-based cryptosystems, such as the Regev and Dual-Regev encryption schemes as well as their leakage-resilient counterparts. The LHL does not hold in the ring setting, when the ring is far from a field, which is typical for efficient cryptosystems. Lyubashevsky et al . (Eurocrypt ’13) proved a “regularity lemma,” which can be used instead of the LHL, but applies only for Gaussian inputs. This is in contrast to the LHL, which applies when the input is drawn from any high min-entropy distribution. Our work presents an approach for generalizing the “regularity lemma” of Lyubashevsky et al . to certain conditional distributions. We assume the input was sampled from a discrete Gaussian distribution and consider the induced distribution, given side-channel leakage on the input. We present three instantiations of our approach, proving that the regularity lemma holds for three natural conditional distributions.

This paper presents an efficient algorithm for computing 11 th -power residue symbols in the cyclo-tomic field ℚ(ζ11), $ \mathbb{Q}\left( {{\zeta }_{11}} \right), $where 11 is a primitive 11 th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7 th -power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.

We introduce a new approach to (deterministic) integer factorisation, which could be described in the cryptographically fashionable term of “factoring with hints”: we prove that, for any ϵ > 0, given the knowledge of the factorisations of O ( N 1/3+ ϵ ) terms surrounding N = pq product of two large primes, we can recover deterministically p and q in O ( N 1/3+ ϵ ) bit operations. This shows that the factorisations of close integers are non trivially related and that consequently one can expect more results along this line of thought.

As one of the most efficient lattice-based signature schemes, and one of the only ones to have seen deployment beyond an academic setting (e.g., as part of the VPN software suite strongSwan), BLISS has attracted a significant amount of attention in terms of its implementation security, and side-channel vulnerabilities of several parts of its signing algorithm have been identified in previous works. In this paper, we present an even simpler timing attack against it. The bimodal Gaussian distribution that BLISS is named after is achieved using a random sign flip during signature generation, and neither the original implementation of BLISS nor strongSwan ensure that this sign flip is carried out in constant time. It is therefore possible to recover the corresponding sign through side-channel leakage (using, e.g., cache attacks or branch tracing). We show that obtaining this single bit of leakage (for a moderate number of signatures) is in fact sufficient for a full key recovery attack. The recovery is carried out using a maximum likelihood estimation on the space of parameters, which can be seen as a statistical manifold. The analysis of the attack thus reduces to the computation of the Fisher information metric.

In this paper we provide a framework for applying classical search and preprocessing to quantum oracles for use with Grover’s quantum search algorithm in order to lower the quantum circuit-complexity of Grover’s algorithm for single-target search problems. This has the effect (for certain problems) of reducing a portion of the polynomial overhead contributed by the implementation cost of quantum oracles and can be used to provide either strict improvements or advantageous trade-offs in circuit-complexity. Our results indicate that it is possible for quantum oracles for certain single-target preimage search problems to reduce the quantum circuit-size from O 2 n / 2 ⋅ m C $O\left(2^{n/2}\cdot mC\right)$ (where C originates from the cost of implementing the quantum oracle) to O ( 2 n / 2 ⋅ m C ) $O(2^{n/2} \cdot m\sqrt{C})$ without the use of quantum ram, whilst also slightly reducing the number of required qubits. This framework captures a previous optimisation of Grover’s algorithm using preprocessing [21] applied to cryptanalysis, providing new asymptotic analysis. We additionally provide insights and asymptotic improvements on recent cryptanalysis [16] of SIKE [14] via Grover’s algorithm, demonstrating that the speedup applies to this attack and impacting upon quantum security estimates [16] incorporated into the SIKE specification [14].