At PKC 2009, May and Ritzenhofen proposed the implicit factorization problem (IFP). They showed that it is undemanding to factor two h-bit RSA moduli N1=p1q1, N2=p2q2 where q1, q2 are both αh-bit, and p1, p2 share uh>2αh the least significant bits (LSBs). Subsequent works mainly focused on extending the IFP to the cases where p1, p2 share some of the most significant bits (MSBs) or the middle bits (MBs). In this paper, we propose a novel generalized IFP where p1 and p2 share an arbitrary number of bit blocks, with each block having a consistent displacement in its position between p1 and p2, and we solve it successfully based on Coppersmith’s method. Specifically, we generate a new set of shift polynomials to construct the lattice and optimize the structure of the lattice by introducing a new variable z=p1. We derive that we can factor the two moduli in polynomial time when u>2(n+1)α(1−α^1/(n+1)) with p1, p2 sharing n blocks. Further, no matter how many blocks are shared, we can theoretically factor the two moduli as long as u>2αln(1/α). In addition, we consider two other cases where the positions of the shared blocks are arbitrary or there are k>2 known moduli. Meanwhile, we provide the corresponding solutions for the two cases. Our work is verified by experiments.

Mixed-Integer Linear Programming (MILP) modeling has become an important tool for both the analysis and the design of symmetric cryptographic primitives. The bit-wise modeling of their nonlinear components, especially the S-boxes, is of particular interest since it allows more informative analysis compared to word-oriented models focusing on counting active S-boxes. At the same time, the size of these models, especially in terms of the number of required inequalities, tends to significantly influence and ultimately limit the applicability of this method to real-world ciphers, especially for larger number of rounds.

It is therefore of great cryptographic significance to study optimal linear inequality descriptions for Boolean functions. The pioneering works of Abdelkhalek et al. (FSE 2017), Boura and Coggia (FSE 2020) and Li and Sun (FSE 2023) provided various heuristic techniques for this computationally hard problem, decomposing it into two algorithmic steps, coined Problem 1 and Problem 2, with the latter being identical to the well-known NP-hard set cover problem, for which there are many heuristic and exact algorithms in the literature.

In this paper, we introduce a novel and efficient branch-and-bound algorithm for generating all minimal, non-redundant candidate inequalities that satisfy a given Boolean function, therefore solving Problem 1 in an optimal manner without relying on heuristics. We furthermore prove that our algorithm correctly computes optimal solutions. Using a number of dedicated optimizations, it provides significantly improved runtimes compared to previous approaches and allows the optimal modeling of the difference distribution tables (DDT) and linear approximation tables (LAT) of many practically used S-boxes. The source code for our algorithm is publicly available as a tool for researchers and practitioners in symmetric cryptography.

We analyze Layered ROLLO-I, a code-based cryptosystem published in IEEE Communications Letters and submitted to the Korean post-quantum cryptography competition. Four versions of Layered ROLLO-I have been proposed in the competition. We show that the first two versions do not provide the claimed security against rank decoding attacks and give reductions to small instances of the original ROLLO-I scheme, which was a candidate in the NIST competition and eliminated there due to rank decoding attacks. As a second contribution, we provide two efficient message recovery attacks, affecting every security level of the first three versions of Layered ROLLO-I and security levels 128 and 192 of the fourth version.

Masking schemes are key in thwarting side-channel attacks due to their robust theoretical foundation. Transitioning from Boolean to arithmetic (B2A) masking is a necessary step in various cryptography schemes, including hash functions, ARX-based ciphers, and lattice-based cryptography. While there exists a significant body of research focusing on B2A software implementations, studies pertaining to hardware implementations are quite limited, with the majority dedicated solely to creating efficient Boolean masked adders. In this paper, we present first- and second-order secure hardware implementations to perform B2A mask conversion efficiently without using masked adder structures. We first introduce a first-order secure low-latency gadget that executes a B2A2k in a single cycle. Furthermore, we propose a second-order secure B2A2k gadget that has a latency of only 4 clock cycles. Both gadgets are independent of the input word size k. We then show how these new primitives lead to improved B2Aq hardware implementations that perform a B2A mask conversion of integers modulo an arbitrary number. Our results show that our new gadgets outperform comparable solutions by more than a magnitude in terms of resource requirements and are at least 3 times faster in terms of latency and throughput. All gadgets have been formally verified and proven secure in the glitch-robust PINI security model. We additionally confirm the security of our gadgets on an FPGA platform using practical TVLA tests.

This work introduces several algorithms related to the computation of orientations in endomorphism rings of supersingular elliptic curves. This problem is at the heart of several results regarding the security of oriented-curves in isogeny-based cryptography. Under the Deuring correspondence, it can be expressed purely in terms of quaternion and boils down to representing integers by ternary quadratic forms. Our main contribution is to show that there exist efficient algorithms to solve this problem for quadratic orders of discriminant $n$ up to $O(p^{4/3})$. Our approach improves upon previous results by increasing this bound from $O(p)$ to $O(p^{4/3})$ and removing some heuristics. We introduce several variants of our new algorithm and provide a careful analysis of their asymptotic running time (without heuristic when it is possible). The best proven asymptotic complexity of one of our variants is $O(n^{3/4}/p)$ in average. The best heuristic variant has a complexity of $O(p^{1/3})$ for big enough $n$. We then introduce several results regarding the computation of ideals between oriented orders. The first application of this is a simplification of the known reduction from vectorization to computing the endomorphism ring, removing the assumption on the factorization of the discriminant. As a second application, we relate the problem of computing fixed-degree isogenies between supersingular curves to the problem of computing orientations in endomorphism rings, and we show that for a large range of degree $d$, our new algorithms improve on the state-of-the-art, and in important special cases, the range of degree $d$ for which there exist a polynomial-time algorithm is increased. In the most special case we consider, when both curves are oriented by a small degree endomorphism, we show heuristically that our techniques allow the computation of isogenies of any degree, assuming they exist.

There has been a notable surge of research on leakage-resilient authenticated encryption (AE) schemes, in the bounded as well as the unbounded leakage model. The latter has garnered significant attention due to its detailed and practical orientation. Designers have commonly utilized (tweakable) block ciphers, exemplified by the TEDT scheme, achieving $\mathcal{O}(n-\log(n^2))$-bit integrity under leakage and comparable AE security in the black-box setting. However, the privacy of TEDT was limited by $n/2$-bits under leakage; TEDT2 sought to overcome these limitations by achieving improved security with $\mathcal{O}(n-\log n)$-bit integrity and privacy under leakage.

This work introduces FEDT, an efficient leakage-resilient authenticated encryption (AE) scheme based on fork-cipher. Compared to the state-of-the-art schemes TEDT and TEDT2, which process messages with a rate of $1/2$ block per primitive call for encryption and one for authentication, FEDT doubles their rates at the price of a different primitive. FEDT employs a more parallelizable tree-based encryption compared to its predecessors while maintaining $\mathcal{O}(n-\log n)$-bit security for both privacy and integrity under leakage. FEDT prioritizes high throughput at the cost of increased latency. For settings where latency is important, we propose FEDT*, which combines the authentication part of FEDT with a CTR-based encryption. FEDT* offers security equivalent to FEDT while increasing the encryption rate of $4/3$ and reducing the latency.

Masking is a prominent strategy to protect cryptographic implementations against side-channel analysis. Its popularity arises from the exponential security gains that can be achieved for (approximately) quadratic resource utilization. Many variants of the countermeasure tailored for different optimization goals have been proposed. The common denominator among all of them is the implicit demand for robust and high entropy randomness. Simply assuming that uniformly distributed random bits are available, without taking the cost of their generation into account, leads to a poor understanding of the efficiency vs. security tradeoff of masked implementations. This is especially relevant in case of hardware masking schemes which are known to consume large amounts of random bits per cycle due to parallelism. Currently, there seems to be no consensus on how to most efficiently derive many pseudo-random bits per clock cycle from an initial seed and with properties suitable for masked hardware implementations. In this work, we evaluate a number of building blocks for this purpose and find that hardware-oriented stream ciphers like Trivium and its reduced-security variant Bivium B outperform most competitors when implemented in an unrolled fashion. Unrolled implementations of these primitives enable the flexible generation of many bits per cycle, which is crucial for satisfying the large randomness demands of state-of-the-art masking schemes. According to our analysis, only Linear Feedback Shift Registers (LFSRs), when also unrolled, are capable of producing long non-repetitive sequences of random-looking bits at a higher rate per cycle for the same or lower cost as Trivium and Bivium B. Yet, these instances do not provide black-box security as they generate only linear outputs. We experimentally demonstrate that using multiple output bits from an LFSR in the same masked implementation can violate probing security and even lead to harmful randomness cancellations. Circumventing these problems, and enabling an independent analysis of randomness generation and masking, requires the use of cryptographically stronger primitives like stream ciphers. As a result of our studies, we provide an evidence-based estimate for the cost of securely generating $n$ fresh random bits per cycle. Depending on the desired level of black-box security and operating frequency, this cost can be as low as $20n$ to $30n$ ASIC gate equivalents (GE) or $3n$ to $4n$ FPGA look-up tables (LUTs), where $n$ is the number of random bits required. Our results demonstrate that the cost per bit is (sometimes significantly) lower than estimated in previous works, incentivizing parallelism whenever exploitable. This provides further motivation to potentially move low randomness usage from a primary to a secondary design goal in hardware masking research.

Traitor tracing schemes [Chor–Fiat–Naor, Crypto ’94] help content distributors fight against piracy and are defined with the content distributor as a trusted authority having access to the secret keys of all users. While the traditional model caters well to its original motivation, its centralized nature makes it unsuitable for many scenarios. For usage among mutually untrusted parties, a notion of *ad hoc* traitor tracing (naturally with the capability of broadcast and revocation) is proposed and studied in this work. Such a scheme allows users in the system to generate their own public/secret key pairs, without trusting any other entity. To encrypt, a list of public keys is used to identify the set of recipients, and decryption is possible with a secret key for any of the public keys in the list. In addition, there is a tracing algorithm that given a list of recipients’ public keys and a pirate decoder capable of decrypting ciphertexts encrypted to them, identifies at least one recipient whose secret key must have been used to construct the said decoder.

Two constructions are presented. The first is based on functional encryption for circuits (conceptually, obfuscation) and has constant-size ciphertext, yet its decryption time is linear in the number of recipients. The second is a generic transformation that reduces decryption time at the cost of increased ciphertext size. A matching lower bound on the trade-off between ciphertext size and decryption time is shown, indicating that the two constructions achieve all possible optimal trade-offs, i.e., they fully demonstrate the Pareto front of efficiency. The lower bound also applies to broadcast encryption (hence all mildly expressive attribute-based encryption schemes) and is of independent interest.

We survey various mathematical tools used in software works multiplying polynomials in \[ \frac{\mathbb{Z}_q[x]}{\left\langle {x^n - \alpha x - \beta} \right\rangle}. \] In particular, we survey implementation works targeting polynomial multiplications in lattice-based cryptosystems Dilithium, Kyber, NTRU, NTRU Prime, and Saber with instruction set architectures/extensions Armv7-M, Armv7E-M, Armv8-A, and AVX2.

There are three emphases in this paper: (i) modular arithmetic, (ii) homomorphisms, and (iii) vectorization. For modular arithmetic, we survey Montgomery, Barrett, and Plantard multiplications. For homomorphisms, we survey (a) various homomorphisms such as Cooley–Tukey FFT, Good–Thomas FFT, Bruun's FFT, Rader's FFT, Karatsuba, and Toom–Cook; (b) various algebraic techniques for adjoining nice properties to the coefficient rings, including localization, Schönhage's FFT, Nussbaumer's FFT, and coefficient ring switching; and (c) various algebraic techniques related to the polynomial moduli, including twisting, composed multiplication, evaluation at $\infty$, truncation, incomplete transformation, striding, and Toeplitz matrix-vector product. For vectorization, we survey the relations between homomorphisms and vector arithmetic.

We then go through several case studies: We compare the implementations of modular multiplications used in Dilithium and Kyber, explain how the matrix-to-vector structure was exploited in Saber, and review the design choices of transformations for NTRU and NTRU Prime with vectorization. Finally, we outline several interesting implementation projects.

Proving whether it is possible to build IND-CCA public-key encryption (PKE) from IND-CPA PKE in a black-box manner is a major open problem in theoretical cryptography. In a significant breakthrough, Gertner, Malkin and Myers showed in 2007 that shielding black-box reductions from IND-CCA to IND-CPA do not exist in the standard model. Shielding means that the decryption algorithm of the IND-CCA scheme does not call the encryption algorithm of the underlying IND-CPA scheme. In other words, it implies that every tentative construction of IND-CCA from IND-CPA must have a re-encryption step when decrypting.

This result was only proven with respect to classical algorithms. In this work we show that it stands in a post-quantum setting. That is, we prove that there is no post-quantum shielding black-box construction of IND-CCA PKE from IND-CPA PKE. In the type of reductions we consider, i.e. post-quantum ones, the constructions are still classical in the sense that the schemes must be computable on classical computers, but the adversaries and the reduction algorithm can be quantum. This suggests that considering quantum notions, which are stronger than their classical counterparts, and allowing for quantum reductions does not make building IND-CCA public-key encryption easier.

Side-channel attacks targeting cryptography may leak only partial or indirect information about the secret keys. There are a variety of techniques in the literature for recovering secret keys from partial information. In this work, we survey several of the main families of partial key recovery algorithms for RSA, (EC)DSA, and (elliptic curve) Diffie-Hellman, the classical public-key cryptosystems in common use today. We categorize the known techniques by the structure of the information that is learned by the attacker, and give simplified examples for each technique to illustrate the underlying ideas.

At CHES 2017, Banik et al. proposed a lightweight block cipher GIFT consisting of two versions GIFT-64 and GIFT-128. Recently, there are lots of authenticated encryption schemes that adopt GIFT-128 as their underlying primitive, such as GIFT-COFB and HyENA. To promote a comprehensive perception of the soundness of the designs, we evaluate their security against differential-linear cryptanalysis.

For this, automatic tools have been developed to search differential-linear approximation for the ciphers based on S-boxes. With the assistance of the automatic tools, we find 13-round differential-linear approximations for GIFT-COFB and HyENA. Based on the distinguishers, 18-round key-recovery attacks are given for the message processing phase and initialization phase of both ciphers. Moreover, the resistance of GIFT-64/128 against differential-linear cryptanalysis is also evaluated. The 12-round and 17-round differential-linear approximations are found for GIFT-64 and GIFT-128 respectively, which lead to 18-round and 19-round key-recovery attacks respectively. Here, we stress that our attacks do not threaten the security of these ciphers.

In recent years, quantum technology has been rapidly developed. As security analyses for symmetric ciphers continue to emerge, many require an evaluation of the resources needed for the quantum circuit implementation of the encryption algorithm. In this regard, we propose the quantum circuit decision problem, which requires us to determine whether there exists a quantum circuit for a given permutation f using M ancilla qubits and no more than K quantum gates within the circuit depth D. Firstly, we investigate heuristic algorithms and classical SAT-based models in previous works, revealing their limitations in solving the problem. Hence, we innovatively propose an improved SAT-based model incorporating three metrics of quantum circuits. The model enables us to find the optimal quantum circuit of an arbitrary 3 or 4-bit S-box under a given optimization goal based on SAT solvers, which has proved the optimality of circuits constructed by the tool, LIGHTER-R. Then, by combining different criteria in the model, we find more compact quantum circuit implementations of S-boxes such as RECTANGLE and GIFT. For GIFT S-box, our model provides the optimal quantum circuit that only requires 8 gates with a depth of 31. Furthermore, our model can be generalized to linear layers and improve the previous SAT-based model proposed by Huang et al. in ASIACRYPT 2022 by adding the criteria on the number of qubits and the circuit depth.

In this work we study algebraic and generic models for group actions, and extend them to the universal composability (UC) framework of Canetti (FOCS 2001). We revisit the constructions of Duman et al. (PKC 2023) integrating the type-safe model by Zhandry (Crypto 2022), adapted to the group action setting, and formally define an algebraic action model (AAM). This model restricts the power of the adversary in a similar fashion to the algebraic group model (AGM). By imposing algebraic behaviour to the adversary and environment of the UC framework, we construct the UC-AAM. Finally, we instantiate UC-AAM with isogeny-based assumptions, in particular the CSIDH action with twists, obtaining the explicit isogeny model, UC-EI; we observe that, under certain assumptions, this model is "closer" to standard UC than the UC-AGM, even though there still exists an important separation. We demonstrate the utility of our definitions by proving UC-EI security for the passive-secure oblivious transfer protocol described by Lai et al. (Eurocrypt 2021), hence providing the first concretely efficient two-message isogeny-based OT protocol in the random oracle model against malicious adversaries.

The problem of Broadcast Encryption (BE) consists in broadcasting an encrypted message to a large number of users or receiving devices in such a way that the emitter of the message can control which of the users can or cannot decrypt it.

Since the early 1990s, the design of BE schemes has received significant interest and many different concepts were proposed. A major breakthrough was achieved by Naor, Naor and Lotspiech (CRYPTO 2001) by partitioning cleverly the set of authorized users and associating a symmetric key to each subset. Since then, while there have been many advances in public-key based BE schemes, mostly based on bilinear maps, little was made on symmetric cryptography.

In this paper, we design a new symmetric-based BE scheme, named $\Sigma\Pi$BE, that relies on logic optimization and consensual security assumptions. It is competitive with the work of Naor et al. and provides a different tradeoff: the bandwidth requirement is significantly lowered at the cost of an increase in the key storage.

There has been a recent interest in proposing quantum protocols whose security relies on weaker computational assumptions than their classical counterparts. Importantly to our work, it has been recently shown that public-key encryption (PKE) from one-way functions (OWF) is possible if we consider quantum public keys. Notice that we do not expect classical PKE from OWF given the impossibility results of Impagliazzo and Rudich (STOC'89).

However, the distribution of quantum public keys is a challenging task. Therefore, the main question that motivates our work is if quantum PKE from OWF is possible if we have classical public keys. Such protocols are impossible if ciphertexts are also classical, given the impossibility result of Austrin et al.(CRYPTO'22) of quantum enhanced key-agreement (KA) with classical communication.

In this paper, we focus on black-box separation for PKE with classical public key and quantum ciphertext from OWF under the polynomial compatibility conjecture, first introduced in Austrin et al.. More precisely, we show the separation when the decryption algorithm of the PKE does not query the OWF. We prove our result by extending the techniques of Austrin et al. and we show an attack for KA in an extended classical communication model where the last message in the protocol can be a quantum state.