## CryptoDB

### Christian Rechberger

#### Publications

Year
Venue
Title
2022
TOSC
We consider SPN schemes, i.e., schemes whose non-linear layer is defined as the parallel application of t ≥ 1 independent S-Boxes over F2n and whose linear layer is defined by the multiplication with a (n · t) × (n · t) matrix over F2. Even if the algebraic representation of a scheme depends on all its components, upper bounds on the growth of the algebraic degree in the literature usually only consider the details of the non-linear layer. Hence a natural question arises: (how) do the details of the linear layer influence the growth of the algebraic degree? We show that the linear layer plays a crucial role in the growth of the algebraic degree and present a new upper bound on the algebraic degree in SP-networks. As main results, we prove that in the case of low-degree round functions with large S-Boxes: (a) an initial exponential growth of the algebraic degree can be followed by a linear growth until the maximum algebraic degree is reached; (b) the rate of the linear growth is proportional to the degree of the linear layer over Ft2n. Besides providing a theoretical insight, our analysis is particularly relevant for assessing the security of the security of cryptographic permutations designed to be competitive in applications like MPC, FHE, SNARKs, and STARKs, including permutations based on the Hades design strategy. We have verified our findings on small-scale instances and we have compared them against the currently best results in the literature, showing a substantial improvement of upper bounds on the algebraic degree in case of low-degree round functions with large S-Boxes.
2021
TOSC
Designing cryptographic permutations and block ciphers using a substitutionpermutation network (SPN) approach where the nonlinear part does not cover the entire state has recently gained attention due to favorable implementation characteristics in various scenarios.For word-oriented partial SPN (P-SPN) schemes with a fixed linear layer, our goal is to better understand how the details of the linear layer affect the security of the construction. In this paper, we derive conditions that allow us to either set up or prevent attacks based on infinitely long truncated differentials with probability 1. Our analysis is rather broad compared to earlier independent work on this problem since we consider (1) both invariant and non-invariant/iterative trails, and (2) trails with and without active S-boxes.For these cases, we provide rigorous sufficient and necessary conditions for the matrix that defines the linear layer to prevent the analyzed attacks. On the practical side, we present a tool that can determine whether a given linear layer is vulnerable based on these results. Furthermore, we propose a sufficient condition for the linear layer that, if satisfied, ensures that no infinitely long truncated differential exists. This condition is related to the degree and the irreducibility of the minimal polynomial of the matrix that defines the linear layer. Besides P-SPN schemes, our observations may also have a crucial impact on the Hades design strategy, which mixes rounds with full S-box layers and rounds with partial S-box layers.
2020
EUROCRYPT
Keyed and unkeyed cryptographic permutations often iterate simple round functions. Substitution-permutation networks (SPNs) are an approach that is popular since the mid 1990s. One of the new directions in the design of these round functions is to reduce the substitution (S-Box) layer from a full one to a partial one, uniformly distributed over all the rounds. LowMC and Zorro are examples of this approach. A relevant freedom in the design space is to allow for a highly non-uniform distribution of S-Boxes. However, choosing rounds that are so different from each other is very rarely done, as it makes security analysis and implementation much harder. We develop the design strategy HADES and an analysis framework for it, which despite this increased complexity allows for security arguments against many classes of attacks, similar to earlier simpler SPNs. The framework builds upon the wide trail design strategy, and it additionally allows for security arguments against algebraic attacks, which are much more of a concern when algebraically simple S-Boxes are used. Subsequently, this is put into practice by concrete instances and benchmarks for a use case that generally benefits from a smaller number of S-Boxes and showcases the diversity of design options we support: A candidate cipher natively working with objects in GF(p), for securing data transfers with distributed databases using secure multiparty computation (MPC). Compared to the currently fastest design MiMC, we observe significant improvements in online bandwidth requirements and throughput with a simultaneous reduction of preprocessing effort, while having a comparable online latency.
2020
ASIACRYPT
Algebraically simple PRFs, ciphers, or cryptographic hash functions are becoming increasingly popular, for example due to their attractive properties for MPC and new proof systems (SNARKs, STARKs, among many others). In this paper, we focus on the algebraically simple construction MiMC, which became an attractive cryptanalytic target due to its simplicity, but also due to its use as a baseline in a competition for more recent algorithms exploring this design space. For the first time, we are able to describe key-recovery attacks on all full-round versions of MiMC over GF(2^n), requiring half the code book. In the chosen-ciphertext scenario, recovering the key from this data for the n-bit full version of MiMC takes the equivalent of less than 2^(n - log_2(n) + 1) calls to MiMC and negligible amounts of memory. The attack procedure is a generalization of higher-order differential cryptanalysis, and it is based on two main ingredients. First, we present a higher-order distinguisher which exploits the fact that the algebraic degree of MiMC grows significantly slower than originally believed. Secondly, we describe an approach to turn this distinguisher into a key-recovery attack without guessing the full subkey. Finally, we show that approximately ceil(log_3(2 * R)) more rounds (where R = ceil(n * log_3(2)) is the current number of rounds of MiMC-n/n) can be necessary and sufficient to restore the security against the key-recovery attack presented here. The attack has been practically verified on toy versions of MiMC. Note that our attack does not affect the security of MiMC over prime fields.
2019
EUROCRYPT
$\textsc {LowMC}$LOWMC is a block cipher family designed in 2015 by Albrecht et al. It is optimized for practical instantiations of multi-party computation, fully homomorphic encryption, and zero-knowledge proofs. $\textsc {LowMC}$LOWMC is used in the $\textsc {Picnic}$PICNIC signature scheme, submitted to NIST’s post-quantum standardization project and is a substantial building block in other novel post-quantum cryptosystems. Many $\textsc {LowMC}$LOWMC instances use a relatively recent design strategy (initiated by Gérard et al. at CHES 2013) of applying the non-linear layer to only a part of the state in each round, where the shortage of non-linear operations is partially compensated by heavy linear algebra. Since the high linear algebra complexity has been a bottleneck in several applications, one of the open questions raised by the designers was to reduce it, without introducing additional non-linear operations (or compromising security).In this paper, we consider $\textsc {LowMC}$LOWMC instances with block size n, partial non-linear layers of size $s \le n$s≤n and r encryption rounds. We redesign LowMC’s linear components in a way that preserves its specification, yet improves LowMC’s performance in essentially every aspect. Most of our optimizations are applicable to all SP-networks with partial non-linear layers and shed new light on this relatively new design methodology.Our main result shows that when $s < n$s<n, each $\textsc {LowMC}$LOWMC instance belongs to a large class of equivalent instances that differ in their linear layers. We then select a representative instance from this class for which encryption (and decryption) can be implemented much more efficiently than for an arbitrary instance. This yields a new encryption algorithm that is equivalent to the standard one, but reduces the evaluation time and storage of the linear layers from $r \cdot n^2$r·n2 bits to about $r \cdot n^2 - (r-1)(n-s)^2$r·n2-(r-1)(n-s)2. Additionally, we reduce the size of LowMC’s round keys and constants and optimize its key schedule and instance generation algorithms. All of these optimizations give substantial improvements for small s and a reasonable choice of r. Finally, we formalize the notion of linear equivalence of block ciphers and prove the optimality of some of our results.Comprehensive benchmarking of our optimizations in various $\textsc {LowMC}$LOWMC applications (such as $\textsc {Picnic}$PICNIC) reveals improvements by factors that typically range between 2x and 40x in runtime and memory consumption.
2019
ASIACRYPT
The block cipher Jarvis and the hash function Friday, both members of the MARVELlous family of cryptographic primitives, are among the first proposed solutions to the problem of designing symmetric-key algorithms suitable for transparent, post-quantum secure zero-knowledge proof systems such as ZK-STARKs. In this paper we describe an algebraic cryptanalysis of Jarvis and Friday and show that the proposed number of rounds is not sufficient to provide adequate security. In Jarvis, the round function is obtained by combining a finite field inversion, a full-degree affine permutation polynomial and a key addition. Yet we show that even though the high degree of the affine polynomial may prevent some algebraic attacks (as claimed by the designers), the particular algebraic properties of the round function make both Jarvis and Friday vulnerable to Gröbner basis attacks. We also consider MiMC, a block cipher similar in structure to Jarvis. However, this cipher proves to be resistant against our proposed attack strategy. Still, our successful cryptanalysis of Jarvis and Friday does illustrate that block cipher designs for “algebraic platforms” such as STARKs, FHE or MPC may be particularly vulnerable to algebraic attacks.
2018
CRYPTO
Recent developments in multi party computation (MPC) and fully homomorphic encryption (FHE) promoted the design and analysis of symmetric cryptographic schemes that minimize multiplications in one way or another. In this paper, we propose with Rastaa design strategy for symmetric encryption that has ANDdepth d and at the same time only needs d ANDs per encrypted bit. Even for very low values of d between 2 and 6 we can give strong evidence that attacks may not exist. This contributes to a better understanding of the limits of what concrete symmetric-key constructions can theoretically achieve with respect to AND-related metrics, and is to the best of our knowledge the first attempt that minimizes both metrics simultaneously. Furthermore, we can give evidence that for choices of d between 4 and 6 the resulting implementation properties may well be competitive by testing our construction in the use-case of removing the large ciphertext-expansion when using the BGV scheme.
2018
TOSC
LowMC is a family of block ciphers designed for a low multiplicative complexity. The specification allows a large variety of instantiations, differing in block size, key size, number of S-boxes applied per round and allowed data complexity. The number of rounds deemed secure is determined by evaluating a number of attack vectors and taking the number of rounds still secure against the best of these. In this paper, we demonstrate that the attacks considered by the designers of LowMC in the version 2 of the round-formular were not sufficient to fend off all possible attacks. In the case of instantiations of LowMC with one of the most useful settings, namely with few applied S-boxes per round and only low allowable data complexities, efficient attacks based on difference enumeration techniques can be constructed. We show that it is most effective to consider tuples of differences instead of simple differences, both to increase the range of the distinguishers and to enable key recovery attacks. All applications for LowMC we are aware of, including signature schemes like Picnic and more recent (ring/group) signature schemes have used version 3 of the roundformular for LowMC, which takes our attack already into account.
2017
EUROCRYPT
2016
ASIACRYPT
2016
TOSC
Recently, many efficient cryptographic hash function design strategies have been explored, not least because of the SHA-3 competition. These designs are, almost exclusively, geared towards high performance on long inputs. However, various applications exist where the performance on short (fixed length) inputs matters more. Such hash functions are the bottleneck in hash-based signature schemes like SPHINCS or XMSS, which is currently under standardization. Secure functions specifically designed for such applications are scarce. We attend to this gap by proposing two short-input hash functions (or rather simply compression functions). By utilizing AES instructions on modern CPUs, our proposals are the fastest on such platforms, reaching throughputs below one cycle per hashed byte even for short inputs, while still having a very low latency of less than 60 cycles. Under the hood, this results comes with several innovations. First, we study whether the number of rounds for our hash functions can be reduced, if only second-preimage resistance (and not collision resistance) is required. The conclusion is: only a little. Second, since their inception, AES-like designs allow for supportive security arguments by means of counting and bounding the number of active S-boxes. However, this ignores powerful attack vectors using truncated differentials, including the powerful rebound attacks. We develop a general tool-based method to include arguments against attack vectors using truncated differentials.
2016
TOSC
We introduce subspace trail cryptanalysis, a generalization of invariant subspace cryptanalysis. With this more generic treatment of subspaces we do no longer rely on specific choices of round constants or subkeys, and the resulting method is as such a potentially more powerful attack vector. Interestingly, subspace trail cryptanalysis in fact includes techniques based on impossible or truncated differentials and integrals as special cases. Choosing AES-128 as the perhaps most studied cipher, we describe distinguishers up to 5-round AES with a single unknown key. We report (and practically verify) competitive key-recovery attacks with very low data-complexity on 2, 3 and 4 rounds of AES. Additionally, we consider AES with a secret S-Box and we present a (generic) technique that allows to directly recover the secret key without finding any information about the secret S-Box. This approach allows to use e.g. truncated differential, impossible differential and integral attacks to find the secret key. Moreover, this technique works also for other AES-like constructions, if some very common conditions on the S-Box and on the MixColumns matrix (or its inverse) hold. As a consequence, such attacks allow to better highlight the security impact of linear mappings inside an AES-like block cipher. Finally, we show that our impossible differential attack on 5 rounds of AES with secret S-Box can be turned into a distinguisher for AES in the same setting as the one recently proposed by Sun, Liu, Guo, Qu and Rijmen at CRYPTO 2016
2015
JOFC
2015
FSE
2015
EUROCRYPT
2014
JOFC
2012
EUROCRYPT
2012
ASIACRYPT
2012
FSE
2011
ASIACRYPT
2010
ASIACRYPT
2010
ASIACRYPT
2009
ASIACRYPT
2009
ASIACRYPT
2009
EUROCRYPT
2009
FSE
2008
FSE
2008
FSE
2008
CRYPTO
2008
CRYPTO
2007
FSE
2006
ASIACRYPT
2006
FSE
2006
FSE

#### Program Committees

Crypto 2022
Crypto 2020
CHES 2019
Eurocrypt 2019
Crypto 2016
FSE 2016
Eurocrypt 2015
FSE 2014 (Program chair)
Asiacrypt 2014
FSE 2013
Crypto 2013
Asiacrypt 2012
FSE 2011
Asiacrypt 2011
FSE 2010
FSE 2009
Eurocrypt 2009