Cryptographic algorithms are the equivalent of locks, seals, security stamps and identification documents over the Internet. They are essential to protect our online bank transactions, credit cards, medical and personal information, and to support e-commerce and e-government. They come in different flavors. Encryption algorithms are necessary to protect sensitive information such as medical data, financial information and Personal Identification Numbers (PINs) from prying eyes. Digital signature algorithms (in combination with hash functions) and MAC algorithms replace hand-written signatures in electronic transactions. Identification protocols allow to securely verify the identity of a remote party. As a whole, cryptology is a research area with a high strategic impact in industry, for individuals, and for society as a whole. The research activity of project-team CASCADE addresses the following topics, which cover most of the areas that are currently active in the international cryptographic community, with a focus on public-key algorithms:

Since the beginning of public-key cryptography, with the seminal Diffie-Hellman paper, many suitable algorithmic problems for cryptography have been proposed and many cryptographic schemes have been designed, together with more or less heuristic proofs of their security relative to the intractability of the underlying problems. However, many of those schemes have thereafter been broken. The simple fact that a cryptographic algorithm withstood cryptanalytic attacks for several years has often been considered as a kind of validation procedure, but schemes may take a long time before being broken. As a consequence, the lack of attacks at some time should never be considered as a full security validation of the proposal.

A completely different paradigm is provided by the concept of “provable” security. A significant line of research has tried to provide proofs in the framework of computational complexity theory (a.k.a. “reductionist” security proofs): the proofs provide reductions from a well-studied problem (factoring, RSA or the discrete logarithm) to an attack against a cryptographic protocol. The techniques are derived from complexity theory, providing (polynomial) reductions. And the more efficient the reduction can be, the better the parameters of the schemes will be.

Unfortunately, in many cases, even just provable security is at the cost of an important loss in terms of efficiency for the cryptographic protocol. Thus, some models have been proposed, trying to deal with the security of efficient schemes: some concrete objects are identified with ideal (or black-box) ones. For example, it is by now usual to identify hash functions with ideal random functions, in the so-called “random-oracle model”. Similarly, block ciphers are identified with families of truly random permutations in the “ideal cipher model”. Another kind of idealization has also been introduced in cryptography, the black-box group, where the group operation, in any algebraic group, is defined by a black-box: a new element necessarily comes from the addition (or the subtraction) of two already known elements. It is by now called the “generic group model”, extended to the bilinear and multi-linear setting. Some works even require several ideal models together to provide some new validations.

But still, such idealization cannot be instantiated in practice, and so one prefers provable security without such idealized assumptions, under new and possibly stronger computational assumptions.
As a consequence, a cryptographer has to deal with the following four important steps, which are all main goals of ours:

The security of almost all public-key cryptographic protocols in use today relies on the presumed hardness of problems from number theory such as factoring and computing discrete logarithms. This is problematic because these problems have very similar underlying structure, and its unforeseen exploit can render all currently used public-key cryptography insecure. This structure was in fact exploited by Shor to construct efficient quantum algorithms that break all hardness assumptions from number theory that are currently in use. And so naturally, an important area of research is to build provably secure protocols based on mathematical problems that are unrelated to factoring and discrete log. One of the most promising directions in this line of research is using lattice problems as a source of computational hardness, which also offer features that other alternative public-key cryptosystems (such as MQ-based, code-based, isogeny-based or hash-based schemes) cannot provide. The ERC Advanced Grant PARQ aims at evaluating the security of lattice-based cryptography, with respect to the most powerful adversaries, such as quantum computers and large-scale parallel computers.

In the meantime, although a universal quantum computer may be some decades in the future, quantum communication and quantum error correcting codes are beginning to become concretely available. It is already possible to prepare, manipulate and precisely control systems involving a few quantum information bits (qubits). Such quantum technologies could help improve the efficiency and security of concrete cryptographic protocols. The ANR JCJC project CryptiQ aims at considering three possible scenarios (first, the simple existence of a quantum attacker, then the access to quantum communication for anyone, and finally a complete quantum world) and studies the consequences on the cryptographic protocols currently available. This implies elaborating adversarial models and designing or analyzing concrete protocols with formal security proofs, in order to get ready as soon as one of these scenarios becomes the new reality.

Fully Homomorphic Encryption (FHE) has become a very active research area since 2009, when IBM announced the discovery of a FHE scheme by Craig Gentry. FHE allows to perform any computation on encrypted data, yielding the result encrypted under the same key. This enables outsourcing computation in the Cloud, on encrypted data, so the Cloud provider does not learn any information. However, FHE does not allow to share the result.

Functional Encryption (FE) is another recent tool that allows an authority to deliver functional decryption keys, for any function

Another approach is to focus on a type of computation over encrypted data of particular interest, namely the ability to search over encrypted data. Here, a client encrypts its data, and sends it to a distant server. The client should then be able to issue queries to the server, asking for elements within the encrypted data that fit some search criterion. The server should be able to correctly answer the query, without learning the client’s data (which remains encrypted), or even the contents of the query (which is also encrypted). In this context, the server is regarded as a honest-but-curious adversary attempting to infer private information as it processes the client’s queries. By restricting the range of functionalities compared to FHE and FE, and allowing a controlled amount of leakage, Searchable Symmetric Encryption (SSE) enables very efficient solutions, which can be deployed at scale.

Cryptographic protocols that are secure when executed in isolation can become completely insecure when multiple such instances are executed concurrently (as is unavoidable on the Internet) or when used as a part of a larger protocol. For instance, a man-in-the-middle attacker participating in two simultaneous executions of a cryptographic protocol might use messages from one of the executions in order to compromise the security of the second – Lowe’s attack on the Needham-Schroeder authentication protocol and Bleichenbacher's attack on SSL work this way. Our research addresses security amidst concurrent executions in secure computation and key exchange protocols.

Secure computation allows several mutually distrustful parties to collaboratively compute a public function of their inputs, while providing the same security guarantees as if a trusted party had performed the computation. Potential applications for secure computation include anonymous voting, privacy-preserving auctions and data-mining. Our recent contributions on this topic include

Many companies have already started the migration to the Cloud and many individuals share their personal informations on social networks.
While some of the data are public information, many of them are personal and even quite sensitive.
Unfortunately, the current access mode is purely right-based: the provider first authenticates the client, and grants him access, or not, according to his rights in the access-control list.
Therefore, the provider itself not only has total access to the data, but also knows which data are accessed, by whom, and how:
privacy, which includes secrecy of data (confidentiality), identities (anonymity), and requests (obliviousness), should be enforced.
Moreover, while high availability can easily be controlled, and thus any defect can immediately be detected, failures in privacy protection can remain hidden for a long time.
The industry of the Cloud introduces a new implicit trust requirement: nobody has any idea at all of where and how his data are stored and manipulated, but everybody should blindly trust the providers.
The providers will definitely do their best, but this is not enough.
Privacy-compliant procedures cannot be left to the responsibility of the provider: however strong the trustfulness of the provider may be, any system or human vulnerability can be exploited against privacy.
This presents too huge a threat to tolerate.
The distribution of the data and the secrecy of the actions must be given back to the users. It requires promoting privacy as a global security notion.

In order to protect the data, one needs to encrypt it. Unfortunately, traditional encryption systems are inadequate for most applications involving big, complex data. Recall that in traditional public key encryption, a party encrypts data to a single known user, which lacks the expressiveness needed for more advanced data sharing. In enterprise settings, a party will want to share data with groups of users based on their credentials. Similarly, individuals want to selectively grant access to their personal data on social networks as well as documents and spreadsheets on Google Docs. Moreover, the access policy may even refer to users who do not exist in the system at the time the data is encrypted. Solving this problem requires an entirely new way of encrypting data.

A first natural approach would be fully homomorphic encryption (FHE, see above), but a second one is also Functional Encryption (FE), that is an emerging paradigm for public-key encryption:
it enables more fine-grained access control to encrypted data, for instance, the ability to specify a decryption policy in the ciphertext so that only individuals who satisfy the policy can decrypt, or the ability to associate keywords to a secret key so that it can only decrypt documents containing the keyword.
Our work on functional encryption centers around two goals:

Another approach is secure multi-party computation protocols, where interactivity might provide privacy in a more efficient way, namely for machine learning techniques.
Machine learning makes an intensive use of comparisons, for the activation of neurons, and new approaches have been proposed for efficient comparisons with interactive protocols.

Searchable Encryption (SE) is another technique that aims to protect users' privacy with regard to data uploaded to the cloud. Searchable Encryption is equally concerned with scalability, with the aim to accomodate large real-world databases. As a concrete application, an email provider may wish to store its users’ emails in an encrypted form to provide privacy; but it is obviously highly desirable that users should still be able to search for emails that contain a given word, or whose date falls within a given range. Businesses may also want to outsource databases containing sensitive information, such as client data, for example to dispense with a costly dedicated IT department. To be usable at all, the outsourced encrypted database should still offer some form of search functionality. Failing that, the entire database must be downloaded to process each query to the database, defeating the purpose of cloud storage.

In many contexts, the amount of data outsourced by a client is large, and the overhead incurred by generic solutions such as FHE or FE becomes prohibitive. The goal of Searchable Encryption is to find practical trade-offs between privacy, functionality, and efficiency. Regarding functionality, the focus is mainly on privately searching over encrypted cloud data, altough many SE schemes also support simple forms of update operation. Regarding privacy, SE typically allows the server to learn some information on the encrypted data.
This information is formally captured by a leakage function. Security proofs show that the cloud server does not learn any more information about the client's data than what is expressed by the leakage function.

The additional flexibility afforded by allowing a controlled amount of leakage enables SE to offer highly efficient solutions, which can be deployed in practice on large datasets. The main goal of our research in this area is to analyze the precise privacy impact of different leakage functions; propose new techniques to reduce this leakage; as well as extend the range of functionality achieved by Searchable Encryption.

In recent years, there has been very significant investment on research on quantum computers – machines that exploit quantum mechanical phenomena to solve mathematical problems that are difficult or intractable for conventional computers. If large-scale quantum computers are ever built, they will be able to break many of the public-key cryptosystems currently in use. This would seriously compromise the confidentiality and integrity of digital communications on the Internet and elsewhere. The goal of post-quantum cryptography (also called quantum-resistant cryptography or quantum-safe cryptography) is to develop cryptographic systems that are secure against both quantum and classical computers, and can interoperate with existing communication protocols and networks.

In 2016, NIST initiated a process to solicit, evaluate, and standardize one or more quantum-resistant public-key cryptographic algorithms. Round 3 candidates were announced on July 22, 2020. Out of the seven finalists, five are based on lattice problems: CRYSTALS-KYBER, NTRU and SABER for encryption, CRYSTALS-DILITHIUM and FALCON for signature. We intend to study the best lattice algorithms in order to assess the security of the five NIST finalists based on the hardness of lattice problems.

With several initiatives such as the development of a 2,000 km quantum network in China, the access of IBM's quantum platform freely available and the efforts made in the EU for instance with the quantum internet alliance team, we can assume that in a further future, not only the adversary has potential access to a quantum computer, but everybody may have access to quantum channels, allowing honest parties to exchange quantum data up to a limited amount. Going one step further than post-quantum cryptography, it is therefore needed to carefully study the security models and properties of classical protocols or the soundness of classical theoretical results in such a setting. Some security notions have already been defined but others have to be extended, such as the formal treatment of superposition attacks initiated by Zhandry.

On the positive side, some quantum primitives which are already well-studied, unconditionally quantum secure and already deployed in practice (such as Quantum Key Distribution) allow for new security properties such as everlasting confidentiality for sensitive long-lived data (which holds even if an attacker stores encrypted data now and decrypts them later when a quantum computer becomes available). We intend to study to what extent allowing honest parties to have access to currently available (or near-term) quantum technologies allows to achieve quantum-enhanced protocols (for classical functionalities) with improved security or efficiency beyond what is possible classically.

Unfortunately, private computation is usually at a huge cost: it definitely costs more to compute on encrypted data than on clear inputs. However, our goal is definitely to reduce this cost, as it will improve the user experience at the same time, with shorter computation time.

Strong privacy for the Cloud would have a huge societal impact since it would revolutionize the trust model: users would be able to make safe use of outsourced storage, namely for personal, financial and medical data, without having to worry about failures or attacks of the server.

Both design of new primitives and study of the best attacks are essential for this goal.

All the results of the team have been published (see the list of publications). They are all related to the research program (see Section ) and the research projects (see Sections and ):

Thomas Vidick, from Caltech, visited Cascade on April 2021.