LPN in Cryptography | EPFL Graph Search

The security of public-key cryptography relies on well-studied hard problems, problems for which we do not have efficient algorithms. Factorization and discrete logarithm are the two most known and used hard problems. Unfortunately, they can be easily solved on a quantum computer by Shor's algorithm. Also, the research area of cryptography demands for crypto-diversity which says that we should offer a range of hard problems for public-key cryptography. If one hard problem proves to be easy, we should be able to provide alternative solutions. Some of the candidates for post-quantum hard problems, i.e. problems which are believed to be hard even on a quantum computer, are the Learning Parity with Noise (LPN), the Learning with Errors (LWE) and the Shortest Vector Problem (SVP). A thorough study of these problems is needed in order to assess their hardness. In this thesis we focus on the algorithmic study of LPN. LPN is a hard problem that is attractive, as it is believed to be post-quantum resistant and suitable for lightweight devices. In practice, it has been employed in several encryption schemes and authentication protocols. At the beginning of this thesis, we take a look at the existing LPN solving algorithms. We provide the theoretical analysis that assesses their complexity. We compare the theoretical results with practice by implementing these algorithms. We study the efficiency of all LPN solving algorithms which allow us to provide secure parameters that can be used in practice. We push further the state of the art by improving the existing algorithms with the help of two new frameworks. In the first framework, we split an LPN solving algorithm into atomic steps. We study their complexity, how they impact the other steps and we construct an algorithm that optimises their use. Given an LPN instance that is characterized by the noise level and the secret size, our algorithm provides the steps to follow in order to solve the instance with optimal complexity. In this way, we can assess if an LPN instance provides the security we require and we show what are the secure instances for the applications that rely on LPN. The second framework handles problems that can be decomposed into steps of equal complexity. Here, we assume that we have an adversary that has access to a finite or infinite number of instances of the same problem. The goal of the adversary is to succeed in just one instance as soon as possible. Our framework provides the strategy that achieves this. We characterize an LPN solving algorithm in this framework and show that we can improve its complexity in the scenario where the adversary is restricted. We show that other problems, like password guessing, can be modeled in the same framework.

Bringing Theory Closer to Practice in Post-quantum and Leakage-resilient Cryptography

Alexandre Raphaël Duc

Modern cryptography pushed forward the need of having provable security. Whereas ancient cryptography was only relying on heuristic assumptions and the secrecy of the designs, nowadays researchers try to make the security of schemes to rely on mathematical problems which are believed hard to solve. When doing these proofs, the capabilities of potential adversaries are modeled formally. For instance, the black-box model assumes that an adversary does not learn anything from the inner-state of a construction. While this assumption makes sense in some practical scenarios, it was shown that one can sometimes learn some information by other means, e.g., by timing how long the computation take. In this thesis, we focus on two different areas of cryptography. In both parts, we take first a theoretical point of view to obtain a result. We try then to adapt our results so that they are easily usable for implementers and for researchers working in practical cryptography. In the first part of this thesis, we take a look at post-quantum cryptography, i.e., at cryptographic primitives that are believed secure even in the case (reasonably big) quantum computers are built. We introduce HELEN, a new public-key cryptosystem based on the hardness of the learning from parity with noise problem (LPN). To make our results more concrete, we suggest some practical instances which make the system easily implementable. As stated above, the design of cryptographic primitives usually relies on some well-studied hard problems. However, to suggest concrete parameters for these primitives, one needs to know the precise complexity of algorithms solving the underlying hard problem. In this thesis, we focus on two recent hard-problems that became very popular in post-quantum cryptography: the learning with error (LWE) and the learning with rounding problem (LWR). We introduce a new algorithm that solves both problems and provide a careful complexity analysis so that these problems can be used to construct practical cryptographic primitives. In the second part, we look at leakage-resilient cryptography which studies adversaries able to get some side-channel information from a cryptographic primitive. In the past, two main disjoint models were considered. The first one, the threshold probing model, assumes that the adversary can put a limited number of probes in a circuit. He then learns all the values going through these probes. This model was used mostly by theoreticians as it allows very elegant and convenient proofs. The second model, the noisy-leakage model, assumes that every component of the circuit leaks but that the observed signal is noisy. Typically, some Gaussian noise is added to it. According to experiments, this model depicts closely the real behaviour of circuits. Hence, this model is cherished by the practical cryptographic community. In this thesis, we show that making a proof in the first model implies a proof in the second model which unifies the two models and reconciles both communities. We then look at this result with a more practical point-of-view. We show how it can help in the process of evaluating the security of a chip based solely on the more standard mutual information metric.

EPFL2015

Implications of Position in Cryptography

Handan Kilinç Alper

In our daily lives, people or devices frequently need to learn their location for many reasons as some services depend on the absolute location or the proximity. The outcomes of positioning systems can have critical effects e.g., on military, emergency. Thus, the security of these systems is quite important. In this thesis, we concentrate on many security aspects of position in cryptography. The first part of this thesis focuses on the theory of distance bounding. A distance bounding protocol is a two-party authentication protocol between a prover and a verifier which considers the distance of the prover as a part of his/her credential. It aims to defeat threats by malicious provers who try to convince that they are closer to the verifier or adversaries which seek to impersonate a far-away prover. In this direction, we first study the optimal security bounds that a distance bounding protocol can achieve. We consider the optimal security bounds when we add some random delays in the distance computation and let the prover involve distance computation. Then, we focus on solving the efficiency problem of public-key distance bounding because the public-key cryptography requires much more computations than the symmetric-key cryptography. We construct two generic protocols (one without privacy, one with) which require fewer computations on the prover side compared to the existing protocols while keeping the highest security level. Then, we describe a new security model involving a tamper-resistant hardware. This model is called the secure hardware model (SHM). We define an all-in-one security model which covers all the threats of distance bounding and an appropriate privacy notion for SHM. The second part of this thesis is to fill the gap between the distance bounding and its real-world applications. We first consider contactless access control. We define an integrated security and privacy model for access control using distance bounding (DB) to defeat relay attacks. We show how a secure DB protocol can be converted to a secure contactless access control protocol. Regarding privacy (i.e., keeping anonymity in a strong sense to an active adversary), we show that the conversion does not always preserve privacy, but it is possible to study it on a case by case basis. Then, we consider contactless payment systems. We design an adversarial model and define formally the contactless payment security against malicious cards and malicious terminals. Accordingly, we design a contactless payment protocol and show its security in our security model. The last part of this thesis focuses on positioning. We consider two problems related to positioning systems: localization and proof of location. In localization, a user aims to find its position by using a wireless network. In proof of location, a user wants to prove his/her position e.g., to have access to a system or authorize itself. We first formally define the problem of localization and construct a formal security model. We describe algorithms and protocols for localization which are secure in our model. Proof of location has been considered formally by Chandran et al. in CRYPTO 2009 and it was proved that achieving security is not possible in the vanilla model. By integrating the localization and the secure hardware model, we obtain a model where we can achieve proof of location.

EPFL2018

Attacks on some post-quantum cryptographic protocols: The case of the Legendre PRF and SIKE

Novak Kaluderovic

Post-quantum cryptography is a branch of cryptography which deals with cryptographic algorithms whose hardness assumptions are not based on problems known to be solvable by a quantum computer, such as the RSA problem, factoring or discrete logarithms.This thesis treats two such algorithms and provides theoretical and practical attacks against them.The first protocol is the generalised Legendre pseudorandom function - a random bit generator computed as the Legendre symbol of the evaluation of a secret polynomial at an element of a finite field. We introduce a new point of view on the protocol by analysing the action of the group of Möbius transformations on the set of secret keys (secret polynomials).We provide a key extraction attack by creating a table which is cubic in the number of the function queries, an improvement over the previous algorithms which only provided a quadratic yield. Furthermore we provide an ever stronger attack for a new set of particularly weak keys.The second protocol that we cover is SIKE - supersingular isogeny key encapsulation.In 2017 the American National Institute of Standards and Technology (NIST) opened a call for standardisation of post-quantum cryptographic algorithms. One of the candidates, currently listed as an alternative key encapsulation candidate in the third round of the standardisation process, is SIKE.We provide three practical side-channel attacks on the 32-bit ARM Cortex-M4 implementation of SIKE.The first attack targets the elliptic curve scalar multiplication, implemented as a three-point ladder in SIKE. The lack of coordinate randomisation is observed, and used to attack the ladder by means of a differential power analysis algorithm.This allows us to extract the full secret key of the target party with only one power trace.The second attack assumes coordinate randomisation is implemented and provides a zero-value attack - the target party is forced to compute the field element zero, which cannot be protected by randomisation. In particular we target both the three-point ladder and isogeny computation in two separate attacks by providing maliciously generated public keys made of elliptic curve points of irregular order.We show that an order-checking countermeasure is effective, but comes at a price of 10% computational overhead. Furthermore we show how to modify the implementation so that it can be protected from all zero-value attacks, i.e., a zero-value is never computed during the execution of the algorithm.Finally, the last attack targets a point swapping procedure which is a subroutine of the three-point ladder. The attack successfully extracts the full secret key with only one power trace even if the implementation is protected with coordinate randomisation or order-checking. We provide an effective countermeasure --- an improved point swapping algorithm which protects the implementation from our attack.

EPFL2022