## IACR News

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Here you can see all recent updates to the IACR webpage. These updates are also available:

#### 07 October 2021

###### Sébastien Canard, Nicolas Desmoulins, Sébastien Hallay, Adel Hamdi, Dominique Le Hello
ePrint Report
The preponderance of smart devices, such as smartphones, has boosted the development and use of mobile applications (apps) in the recent years. This prevalence induces a large volume of mobile app usage data. The analysis of such information could lead to a better understanding of users' behaviours in using the apps they have installed, even more if these data can be coupled with a given context (location, time, date, sociological data...). However, mobile and apps usage data are very sensitive, and are today considered as personal. Their collection and use pose serious concerns associated with individuals' privacy. To reconcile harnessing of data and privacy of users, we investigate in this paper the possibility to conduct privacy-preserving mobile data usage statistics that will prevent any inference or re-identification risks. The key idea is for each user to encrypt their (private and sensitive) inputs before sending them to the data processor. The possibility to perform statistics on those data is then possible thanks to the use of functional encryption, a cryptographic building block permitting to perform some allowed operations over encrypted data. In this paper, we first show how it is possible to obtain such individuals' usage of their apps, which step is necessary for our use case, but can at the same time pose some security problems w.r.t. those apps. We then design our new encryption scheme, adding some fault tolerance property to a recent dynamic decentralized function encryption scheme. We finally show how we have implemented all that, and give some benchmarks.
###### Subhadeep Banik, Khashayar Barooti, Serge Vaudenay, Hailun Yan
ePrint Report
Cryptanalysis of the LowMC block cipher when the attacker has access to a single known plaintext/ciphertext pair is a mathematically challenging problem. This is because the attacker is unable to employ most of the standard techniques in symmetric cryptography like linear and differential cryptanalysis. This scenario is particularly relevant while arguing the security of the \picnic digital signature scheme in which the plaintext/ciphertext pair generated by the LowMC block cipher serves as the public (verification) key and the corresponding LowMC encryption key also serves as the secret (signing) key of the signature scheme. In the paper by Banik et al. (IACR ToSC 2020:4), the authors used a linearization technique of the LowMC S-box to mount attacks on some instances of the block cipher. In this paper, we first make a more precise complexity analysis of the linearization attack. Then, we show how to perform a 2-stage MITM attack on LowMC. The first stage reduces the key candidates corresponding to a fraction of key bits of the master key. The second MITM stage between this reduced candidate set and the remaining fraction of key bits successfully recovers the master key. We show that the combined computational complexity of both these stages is significantly lower than those reported in the ToSC paper by Banik et al.
###### Jan Richter-Brockmann, Ming-Shing Chen, Santosh Ghosh, Tim Güneysu
ePrint Report
BIKE is a Key Encapsulation Mechanism selected as an alternate candidate in NIST’s PQC standardization process, in which performance plays a signiﬁcant role in the third round. This paper presents FPGA implementations of BIKE with the best area-time performance reported in literature. We optimize two key arithmetic operations, which are the sparse polynomial multiplication and the polynomial inversion. Our sparse multiplier achieves time-constancy for sparse polynomials of indeﬁnite Hamming weight used in BIKE’s encapsulation. The polynomial inversion is based on the extended Euclidean algorithm, which is unprecedented in current BIKE implementations. Our optimized design results in a 5.5 times faster key generation compared to previous implementations based on Fermat’s little theorem.

Besides the arithmetic optimizations, we present a united hardware design of BIKE with shared resources and shared sub-modules among KEM functionalities. On Xilinx Artix-7 FPGAs, our light-weight implementation consumes only 3 777 slices and performs a key generation, encapsulation, and decapsulation in 3 797 µs, 443 µs, and 6 896 µs, respectively. Our high-speed design requires 7 332 slices and performs the three KEM operations in 1 672 µs, 132 µs, and 1 892 µs, respectively.
###### Hanlin Liu, Yu Yu
ePrint Report
Blum, Kalai and Wasserman (JACM 2003) gave the first sub-exponential algorithm to solve the Learning Parity with Noise (LPN) problem. In particular, consider the LPN problem with constant noise $\mu=(1-\gamma)/2$. The BKW solves it with space complexity $2^{\frac{(1+\epsilon)n}{\log n}}$ and time/sample complexity $2^{\frac{(1+\epsilon)n}{\log n}}\cdot 2^{O(n^{\frac{1}{1+\epsilon}})}$ for small constant $\epsilon\to 0^+$. We propose a variant of the BKW by tweaking Wagner's generalized birthday problem (Crypto 2002) and adapting the technique to a $c$-ary tree structure. In summary, our algorithm achieves the following:

(Time-space tradeoff). We obtain the same time-space tradeoffs for LPN and LWE as those given by Esser et al. (Crypto 2018), but without resorting to any heuristics. For any $2\leq c\in\mathbb{N}$, our algorithm solves the LPN problem with time/sample complexity $2^{\frac{\log c(1+\epsilon)n}{\log n}}\cdot 2^{O(n^{\frac{1}{1+\epsilon}})}$ and space complexity $2^{\frac{\log c(1+\epsilon)n}{(c-1)\log n}}$, where one can use Grover's quantum algorithm or Dinur et al.'s dissection technique (Crypto 2012) to further accelerate/optimize the time complexity.

(Time/sample optimization). A further adjusted variant of our algorithm solves the LPN problem with sample, time and space complexities all kept at $2^{\frac{(1+\epsilon)n}{\log n}}$ for $\epsilon\to 0^+$, saving factor $2^{\Omega(n^{\frac{1}{1+\epsilon}})}$ in time/sample compared to the original BKW, and the variant of Devadas et al. (TCC 2017). This benefits from a careful analysis of the error distribution among the correlated candidates, and therefore avoids repeating the same process $2^{\Omega(n^{\frac{1}{1+\epsilon}})}$ times on fresh new samples.

(Sample reduction) Our algorithm provides an alternative to Lyubashevsky's BKW variant (RANDOM 2005) for LPN with a restricted amount of samples. In particular, given $Q=n^{1+\epsilon}$ (resp., $Q=2^{n^{\epsilon}}$) samples, our algorithm saves a factor of $2^{\Omega(n)/(\log n)^{1-\kappa}}$ (resp., $2^{\Omega(n^{\kappa})}$) for constant $\kappa \to 1^-$ in running time while consuming roughly the same space, compared with Lyubashevsky's algorithm.

We seek to bridge the gaps between theoretical and heuristic LPN solvers, but take a different approach from Devadas et al. (TCC 2017). We exploit weak yet sufficient conditions (e.g., pairwise independence), and the analysis uses only elementary tools (e.g., Chebyshev's inequality).
###### Dan Boneh, Wilson Nguyen, Alex Ozdemir
ePrint Report
We construct efficient functional commitments for all bounded size arithmetic circuits. A (function hiding) functional commitment scheme allows a committer to commit to a secret function f and later prove that y = f(x) for public x and y—without revealing any other information about f. Thus, functional commitments allow the operator of a secret process to prove that the process is being applied uniformly to everyone. Possible applications include bail decisions, credit scores, online ranking algorithms, and proprietary software-as-a-service. To build functional commitments, we introduce a new type of protocol: a proof of function relation (PFR) to show that a committed relation is a function. We show that combining a suitable preprocessing zk-SNARK with a PFR yields a secure functional commitment scheme. We then construct efficient PFRs for two popular preprocessing zk-SNARKs, and obtain two functional commitment schemes for arithmetic circuits. These constructions build on polynomial commitments (a special case of functional commitments), so our work shows that polynomial commitments are “complete” for functional commitments.

#### 06 October 2021

###### IRIF, Université de Paris, Paris, France
Job Posting
IRIF (Institut de Recherche en Informatique Fondamentale, https://www.irif.fr/) will be offering one to two years of postdoc in cryptography, starting October 2022. Topics of interest include, but are not limited to, secure computation, zero-knowledge proofs, post-quantum cryptography, code-based cryptography, and foundational aspects of cryptography (including black-box separations and connections to learning theory). Anyone interested should apply via algocomp-apply (at) irif (dot) fr (see https://www.irif.fr/postes/postdoc for how to apply). The deadline is November 1st, 2021. Further details: IRIF (research institute in theoretical computer science) is a laboratory of the University of Paris. It's the largest TCS lab in France, with more than 90 permanent members and dozens of PhDs & postdocs. It is located in the south of Paris, and is well connected to everything via public transportation.

Closing date for applications:

Contact: Geoffroy Couteau

###### Microsoft Research, Redmond, WA
Job Posting
Research Internships at Microsoft provide a dynamic environment for research careers with a network of world-class research labs led by globally-recognized scientists and engineers. Our researchers and engineers pursue innovation in a range of scientific and technical disciplines to help solve complex challenges in diverse fields, including computing, healthcare, economics, and the environment.

Digital identities are foundational to the modern web and to the many services enabled by various cloud providers. The Cryptography and Privacy Research Group at Microsoft Research, Redmond, is creating new privacy and transparency technologies for the digital identity ecosystem, with the goal of giving users more control over their identity and visibility into its usage. We are looking for a research intern for the spring of 2022 to work with us on privacy-preserving and auditable data structures and applications to future digital identity ecosystems.

Closing date for applications:

Contact: Kim Laine (kim.laine@microsoft.com) or Esha Ghosh (esha.ghosh@microsoft.com)

###### The University of Manchester, Department of Computer Science, Manchester, UK
Job Posting

We are looking for a Research Associate (PostDoc) in Secure & Privacy-preserving AI Models to join our ambitious EnnCore (https://enncore.github.io/) project.

You will enjoy designing, developing and evaluating novel AI models (deep neural networks) that are privacy-preserving and robust against attacks. The project will involve the continuous interaction with experts in explainable AI and formal software verification. You will also have the opportunity to build use cases and to collaborate with domain experts in areas such as cancer research and energy trading. You will design, develop and evaluate new models in the context of their accuracy, privacy-protection and robustness. This position may include research on a diverse set of techniques such as federated learning, homomorphic encryption, multiparty computation and adversarial methods. The post is initially for one year, with the possibility for extensions.

You should have a PhD or equivalent in Computer Science or a closely related field together with a track record of international publications in applied machine learning or secure computation. Examples of fields of interests are:

(1) Federated Learning
(2) Homomorphic Encryption
(3) Secure Multiparty Computation
(4) Differential Privacy
(5) Safety Mechanisms in AI Systems

Closing date for applications:

Contact: Dr Mustafa A. Mustafa
mustafa.mustafa@manchester.ac.uk

###### Heliax, Anoma Protocol, Remote
Job Posting
Heliax is looking for a cryptographer & researcher interested in zero-knowledge cryptographic protocols and their application to distributed ledger technology to work with us to design, evaluate, and implement zero-knowledge proof constructions such as zkSNARKs and zkSTARKs, distributed cryptographic protocols such as threshold encryption and distributed key generation, and cryptographic primitives such as elliptic curves and hash functions, then put this cryptography into practice in order to realise privacy and scalability capabilities required by the next generation of blockchain networks. This role offers the chance to work closely with a small team on compelling cross-disciplinary problems in theoretical computer science, cryptography, game theory, economics, and systems design, and enjoy a high degree of independence in working conditions and task prioritization.

Closing date for applications:

Contact: jobs@heliax.dev

###### New Jersey Institute of Technology (NJIT), USA
Job Posting
Shantanu Sharma, Assistant Professor @ Computer Science at NJIT, has multiple fully funded Ph.D. positions in the area of databases, secure data processing, IoT, cloud/edge computing, blockchain, and secure model learning. Webpage: https://web.njit.edu/~ss797/

Details: NJIT is a Rank 1 Research University, situated in New York Metropolitan area, and is about 7 miles away from the beautiful New York City. New York Metropolitan area is a key part of the US and is the hub of several major tech and research companies. The qualified candidates will have opportunities for research internships and joint projects with lead-industrial companies.
The position is looking for highly motivated graduate as well as undergraduate students to explore, design, and implement algorithms for databases, secure computing, IoT, blockchain, and machine learning. Topics are as follows:
1. GDPR compliance trustworthy database systems
a. The implication of privacy laws in database systems
b. Design of new modules for GDPR compliance secure databases
2. Trustworthy data-driven systems
a. Application of blockchains in data-driven systems
b. Applications of security techniques in data-driven system pipeline
3. Scalable and trustworthy database processing
a. Use of encryption and secret-sharing techniques
b. Use of secure hardware
Outcome: The work will expose the student to novel data management algorithms, programming with secure hardware (Intel SGX), cluster computing frameworks such as MapReduce and Spark, and the basics of secure computing using cryptographic techniques.

Requirements:
1. You will need to be highly enthusiastic, learner, and also be interested in coding/prototyping algorithms and systems
2. Adequate knowledge of algorithms, programming, and relational database systems
3. Knowledge of Java, SQL, and C/C++
4. You must be an Undergraduate/Master student in computer science or a related field

Closing date for applications:

Contact: Shantanu Sharma shantanu[DOT]sharma[AT]njit[DOT]edu Please send your CV and other information (e.g., GitHub account, sample projects, etc.) to: Shantanu Sharma

###### University of Stuttgart, Institute of Information Security
Job Posting
The Institute of Information Security at University of Stuttgart offers a

fully-funded Postdoc position in formal verification.

The successful candidate is expected to work on tool-supported formal verification of security-critical systems and security protocols.

The position is available immediately with an internationally competitive salary (German public salary scale TV-L E13 or TV-L E14, depending on the candidate's qualification, ranging from about 4.600 Euro to 6.200 Euro monthly gross salary). The appointment period follows the German Wissenschaftszeitvertragsgesetz (WissZeitVg), ranging from one year to up to six years.

The Institute of Information Security offers a creative international environment for top-level international research in Germany's high-tech region.

The successful candidate should have a Ph.D. (or should be very close to completion thereof) in Computer Science, Mathematics, Information Security, or a related field. We value strong analytical skills and

• solid knowledge of logic, proofs and/or formal verification techniques (Theorem Proving, Type Checking, etc.),
• solid programming experience.

Knowledge in security is not required, but a plus. Knowledge of German is not required.

See https://www.sec.uni-stuttgart.de/institute/job-openings/ for details of how to apply.

October 31st, 2021.

Late applications will be considered until the position is filled.

Closing date for applications:

Contact: Prof. Dr. Ralf Küsters
University of Stuttgart
Institute of Information Security
ralf.kuesters@sec.uni-stuttgart.de

#### 05 October 2021

###### Thomas Agrikola, Geoffroy Couteau, Sven Maier
ePrint Report
The goal of anonymous whistleblowing is to publicly disclose a message while at the same time hiding the identity of the sender in a way that even if suspected of being the sender, this cannot be proven. While many solutions to this problem have been proposed over the years, they all require some form of interaction with trusted or non-colluding parties. In this work, we ask whether this is fundamentally inherent. We put forth the notion of anonymous transfer as a primitive allowing to solve this problem without relying on any participating trusted parties.

We initiate the theoretical study of this question, and derive negative and positive results on the existence of such a protocol. We refute the feasibility of asymptotically secure anonymous transfer, where the message will be received with overwhelming probability while at the same time the identity of the sender remains hidden with overwhelming probability. On the other hand, resorting to fine-grained cryptography, we provide a heuristic instantiation (assuming ideal obfuscation) which guarantees that the message will be correctly received with overwhelming probability and the identity of the sender leaks with vanishing probability. Our results provide strong foundations for the study of the possibility of anonymous communications through authenticated channels, an intriguing goal which we believe to be of fundamental interest.
###### Eik List
ePrint Report
Leakage-resilient authenticated encryption (AE) schemes received considerable attention during the previous decade. Two core security models of bounded and unbounded leakage have evolved, where the latter has been motivated in a very detailed and practice-oriented manner. In that setting, designers often build schemes based on (tweakable) block ciphers due to the small state size, such as the recent two-pass AE scheme TEDT from TCHES 1/2020. TEDT is interesting due to its high security guarantees of O(n - log(n^2))-bit integrity under leakage and similar AE security in the black-box setting. Though, a detail limited it to provide only n/2-bit privacy under leakage. In this work, we extend TEDT to TEDT2 in three aspects with the help of a tweakable block cipher with a 3n-bit tweakey: we (1) adopt the idea from the design team of Romulus of replacing TEDT's previous internal hash function with Naito's MDPH, (2) move the nonce from the hash to the tag-generation function both for more efficiency, and (3) strengthen the security of the encryption to obtain beyond-birthday-bound security also under leakage.
###### Luk Bettale, Simon Montoya, Guénaël Renault
ePrint Report
The NIST selection process for standardizing Post-Quantum Cryptography Mechanisms is currently running. Many papers already studied their theoretical security, but the resistance in deployed device has not been much investigated so far. In particular, fault attack is a serious threat for algorithms implemented in embedded devices. One particularly powerful technique is to use safe-error attacks. Such attacks exploit the fact that a specific fault may or may not lead to a faulty output depending on a secret value. In this paper, we investigate the resistance of various Post-Quantum candidates algorithms against such attacks.
###### Dongxi Liu
ePrint Report
We propose a new hard problem, called the Embedded Multilayer Equations (eMLE) problem in this paper. An example of eMLE, with one secret variable x and three layers, is given below.

6268 = 57240 * x + (1248 * x + (9 * x mod 16) mod 2053) mod 65699

In this example, the eMLE problem is to find x from the above equation. eMLE in this paper has the same number of variables and equations. The hardness of eMLE problem lies in its layered structure. Without knowing the eMLE value of lower layer (i.e., the layer with modulus 2053), the top layer (i.e., the layer with modulus 65699) has many candidate solutions; the adversary has to search the solution space for a few valid ones. A lower-bound for the number of searches has been proven in the paper, together with the expected number of valid solutions. The hardness of eMLE can be increased by adding more layers, without changing the number of variables and equations; no existing NP-complete problems have this feature. Over the hardness of eMLE, a post-quantum signature scheme, eMLE-Sig, is constructed. Compared with all existing signature schemes (conventional and post-quantum), eMLE-Sig might be the simplest to understand, analyze, instantiate, and implement. At the security level above 128 bits, five configurations are provided; all of them have keys and signatures smaller than RSA keys and signatures (above 380 bytes) at the 128-bit security level. The smallest configuration is with two variables and three layers, having 84.1/52.2 bytes for private/public key and 168.4 bytes for signatures.
###### Zeyu Liu, Daniele Micciancio, Yuriy Polyakov
ePrint Report
A comparison of two encrypted numbers is an important operation needed in many machine learning applications, for example, decision tree or neural network inference/training. An efficient instantiation of this operation in the context of fully homomorphic encryption (FHE) can be challenging, especially when a relatively high precision is sought. The conventional FHE way of evaluating the comparison operation, which is based on the sign function evaluation using FHEW/TFHE bootstrapping, can only support very small precision (practically limited to 5 bits or so). For higher precision, the runtime complexity scales linearly with the ciphertext (plaintext) modulus (i.e., exponentially with the modulus bit size). We propose sign function evaluation algorithms that scale logarithmically with the ciphertext (plaintext) modulus, enabling the support of large-precision comparison in practice. Our sign evaluation algorithms are based on an iterative use of homomorphic floor function algorithms, which are also derived in our work. Further, we generalize our procedures for floor function evaluation to arbitrary function evaluation, which can be used to support both small plaintext moduli (directly) and larger plaintext moduli (by using a homomorphic digit decomposition algorithm, also suggested in our work). We implement all these algorithms using the PALISADE lattice cryptography library, introducing several implementation-specific optimizations along the way, and discuss our experimental results.
###### Dakshita Khurana, Akshayaram Srinivasan
ePrint Report
Recent exciting breakthroughs, starting with the work of Chattopadhyay and Zuckerman (STOC 2016) have achieved the first two-source extractors that operate in the low min-entropy regime. Unfortunately, these constructions suffer from non-negligible error, and reducing the error to negligible remains an important open problem. In recent work, Garg, Kalai, and Khurana (GKK, Eurocrypt 2020) investigated a meaningful relaxation of this problem to the computational setting, in the presence of a common random string (CRS). In this relaxed model, their work built explicit two-source extractors for a restricted class of unbalanced sources with min-entropy $n^{\gamma}$ (for some constant $\gamma$) and negligible error, under the sub-exponential DDH assumption.

In this work, we investigate whether computational extractors in the CRS model be applied to more challenging environments. Specifically, we study network extractor protocols (Kalai et al., FOCS 2008) and extractors for adversarial sources (Chattopadhyay et al., STOC 2020) in the CRS model. We observe that these settings require extractors that work well for balanced sources, making the GKK results inapplicable. We remedy this situation by obtaining the following results, all of which are in the CRS model and assume the sub-exponential hardness of DDH.

- We obtain optimal'' computational two-source and non-malleable extractors for balanced sources: requiring both sources to have only poly-logarithmic min-entropy, and achieving negligible error. To obtain this result, we perform a tighter and arguably simpler analysis of the GKK extractor. - We obtain a single-round network extractor protocol for poly-logarithmic min-entropy sources that tolerates an optimal number of adversarial corruptions. Prior work in the information-theoretic setting required sources with high min-entropy rates, and in the computational setting had round complexity that grew with the number of parties, required sources with linear min-entropy, and relied on exponential hardness (albeit without a CRS). - We obtain an optimal'' {\em adversarial source extractor} for poly-logarithmic min-entropy sources, where the number of honest sources is only 2 and each corrupted source can depend on either one of the honest sources. Prior work in the information-theoretic setting had to assume a large number of honest sources.
###### Ilia Iliashenko, Christophe Nègre, Vincent Zucca
ePrint Report
Fully Homomorphic Encryption (FHE) gives the ability to evaluate any function over encrypted data. However, despite numerous improvements during the last decade, the computational overhead caused by homomorphic computations is still very important. As a consequence, optimizing the way of performing the computations homomorphically remains fundamental. Several popular FHE schemes such as BGV and BFV encode their data, and thus perform their computations, in finite fields. In this work, we study and exploit algebraic relations occurring in prime characteristic allowing to speed-up the homomorphic evaluation of several functions over prime fields.

More specifically we give several examples of unary functions: "modulo", "is power of $b$" and "Hamming weight" and "Mod2'" whose homomorphic evaluation complexity over $\mathbb{F}_p$ can be reduced from the generic bound $\sqrt{2p} + \mathcal{O}(\log(p))$ homomorphic multiplications, to $\sqrt{p} + \mathcal{O}(\log(p))$, $\mathcal{O}(\log (p))$, $\mathcal{O}(\sqrt{p/\log (p)})$ and $\mathcal{O}(\sqrt{p/(\log (p))})$ respectively. Additionally we provide a proof of a recent claim regarding the structure of the polynomial interpolation of the "less-than" bivariate function which confirms that this function can be evaluated in $2p-6$ homomorphic multiplications instead of $3p-5$ over $\mathbb{F}_p$ for $p\geq 5$.
###### Aayush Jain, Huijia Lin, Amit Sahai
ePrint Report
In this work, we study what minimal sets of assumptions suffice for constructing indistinguishability obfuscation ($i\mathcal{O}$). We prove:

{\bf Theorem}(Informal): Assume sub-exponential security of the following assumptions:

- the Learning Parity with Noise ($\mathsf{LPN}$) assumption over general prime fields $\mathbb{F}_p$ with polynomially many $\mathsf{LPN}$ samples and error rate $1/k^\delta$, where $k$ is the dimension of the $\mathsf{LPN}$ secret, and $\delta>0$ is any constant;

- the existence of a Boolean Pseudo-Random Generator ($\mathsf{PRG}$) in $\mathsf{NC}^0$ with stretch $n^{1+\tau}$, where $n$ is the length of the $\mathsf{PRG}$ seed, and $\tau>0$ is any constant;

- the Decision Linear ($\mathsf{DLIN}$) assumption on symmetric bilinear groups of prime order.

Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists. Further, assuming only polynomial security of the aforementioned assumptions, there exists collusion resistant public-key functional encryption for all polynomial-size circuits.}

This removes the reliance on the Learning With Errors (LWE) assumption from the recent work of [Jain, Lin, Sahai STOC'21]. As a consequence, we obtain the first fully homomorphic encryption scheme that does not rely on any lattice-based hardness assumption. Our techniques feature a new notion of randomized encoding called Preprocessing Randomized Encoding (PRE) that, essentially, can be computed in the exponent of pairing groups. When combined with other new techniques, PRE gives a much more streamlined construction of $\iO$ while still maintaining reliance only on well-studied assumptions.
###### Thomas Pornin
ePrint Report
Lossless compression algorithms such as DEFLATE strive to reliably process arbitrary inputs, while achieving compressed sizes as low as possible for commonly encountered data inputs. It is well-known that it is mathematically impossible for a compression algorithm to simultaneously achieve non-trivial compression on some inputs (i.e. compress these inputs into strictly shorter outputs) and to never expand any other input (i.e. guaranteeing that all inputs will be compressed into an output which is no longer than the input); this is a direct application of the "pigeonhole principle". Despite their mathematical impossibility, we show in this paper how to build such paradoxical compression and decompression algorithms, with the aid of some tools from cryptography, notably verifiable delay functions, and, of course, by slightly cheating.