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A Lattice-Based Traitor Tracing Scheme

San Ling, Damien Stehlé

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

A traitor tracing scheme is a multi-receiver encryption scheme where malicious receiver coalitions aiming at building pirate decryption devices are deterred by the existence of a tracing algorithm: Using the pirate decryption device, the tracing algorithm can recover at least one member of the malicious coalition. All existing traitor tracing schemes rely either on rather inefficient generic constructions from arbitrary encryption schemes and collusion-secure fingerprinting codes, or on algebraic constructions exploiting the assumed hardness of variants of the Discrete Logarithm Problem. In this work, we present the first algebraic construction of a traitor tracing encryption scheme whose security relies on the assumed (quantum) worst-case hardness of standard lattice problems. The scheme is public-key, provably resists Chosen Plaintext Attacks and allows for minimal access black-box tracing (i.e., tracing works even if granted a very limited access to the pirate decryption device). It inherits the standard features of lattice-based cryptography, such as provable security under mild computational assumptions, conjectured resistance to quantum computers, and asymptotic efficiency. For proving the security, we introduce a Learning With Errors variant of the k-SIS problem from Boneh and Freeman [PKC'11], which we prove at least as hard as the standard LWE problem. We also describe a variant of our scheme with security based on the assumed hardness of the Ring Learning With Errors problem which achieves quasi-optimal asymptotic performance with respect to the security parameter.

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A traitor tracing scheme is a multi-receiver encryption scheme where malicious receiver coalitions aiming at building pirate decryption devices are deterred by the existence of a tracing algorithm: Using the pirate decryption device, the tracing algorithm can recover at least one member of the malicious coalition. All existing traitor tracing schemes rely either on rather inefficient generic constructions from arbitrary encryption schemes and collusion-secure fingerprinting codes, or on algebraic constructions exploiting the assumed hardness of variants of the Discrete Logarithm Problem. In this work, we present the first algebraic construction of a traitor tracing encryption scheme whose security relies on the assumed (quantum) worst-case hardness of standard lattice problems. The scheme is public-key, provably resists Chosen Plaintext Attacks and allows for minimal access black-box tracing (i.e., tracing works even if granted a very limited access to the pirate decryption device). It inherits the standard features of lattice-based cryptography, such as provable security under mild computational assumptions, conjectured resistance to quantum computers, and asymptotic efficiency. For proving the security, we introduce a Learning With Errors variant of the k-SIS problem from Boneh and Freeman [PKC'11], which we prove at least as hard as the standard LWE problem. We also describe a variant of our scheme with security based on the assumed hardness of the Ring Learning With Errors problem which achieves quasi-optimal asymptotic performance with respect to the security parameter.
Lattice-based cryptography is a branch of cryptography exploiting the presumed hardness of some well-known problems on lattices. Its main advantages are its simplicity, efficiency, and apparent security against quantum computers. The principle of the security proofs in lattice-based cryptography is to show that attacking a given scheme is at least as hard as solving a particular problem, as the Learning with Errors problem (LWE) or the Small Integer Solution problem (SIS). Then, by showing that those two problems are at least as hard to solve than a hard problem on lattices, presumed polynomial time intractable, we conclude that the constructed scheme is secure.In this thesis, we improve the foundation of the security proofs and build new cryptographic schemes. We study the hardness of the SIS and LWE problems, and of some of their variants on integer rings of cyclotomic fields and on modules on those rings. We show that there is a classical hardness proof for the LWE problem (Regev's prior reduction was quantum), and that the module variants of SIS and LWE are also hard to solve. We also give two new lattice-based group signature schemes, with security based on SIS and LWE. One is the first lattice-based group signature with logarithmic signature size in the number of users. And the other construction allows another functionality, verifier-local revocation. Finally, we improve the size of some parameters in the work on cryptographic multilinear maps of Garg, Gentry and Halevi in 2013.
The growing reliance on cloud computing has brought significant attention to the security and privacy of outsourced data. Subset Predicate Encryption (SPE) provides fine-grained access control suitable for cloud environments, enabling selective decryption based on predicate evaluation. However, most existing SPE constructions are vulnerable to quantum attacks and fail to address the critical issue of key leakage or user collusion. In this work, we present a lattice-based SPE scheme that is post-quantum secure and augmented with traitor tracing functionality. By integrating fingerprinting codes into the encryption and key generation processes, our design enables the identification of malicious users responsible for leaked decryption capabilities. Our scheme offers strong resistance against quantum adversaries and practical traceability in cloud-based data sharing scenarios.
Lattice-based cryptography is the use of conjectured hard problems on point lattices in ℝn as the foundation for secure cryptographic systems. Attractive features of lattice cryptography include apparent resistance to quantum attacks (in contrast with most number-theoretic cryptography), high asymptotic efficiency and parallelism, security under worst-case intractability assumptions, and solutions to long-standing open problems in cryptography. This work surveys most of the major developments in lattice cryptography over the past ten years. The main focus is on the foundational short integer solution (SIS) and learning with errors (LWE) problems (and their more efficient ring-based variants), their provable hardness assuming the worst-case intractability of standard lattice problems, and their many cryptographic applications.
We describe public key encryption schemes with security provably based on the worst case hardness of the approximate Shortest Vector Problem in some structured lattices, called ideal lattices. Under the assumption that the latter is exponentially hard to solve even with a quantum computer, we achieve CPA-security against subexponential at- tacks, with (quasi-)optimal asymptotic performance: if n is the security parameter, both keys are of bit-length O(n) and the amortized costs of both encryption and decryption are O(1) per message bit. Our construc- tion adapts the trapdoor one-way function of Gentry et al. (STOC'08), based on the Learning With Errors problem, to structured lattices. Our main technical tools are an adaptation of Ajtai's trapdoor key genera- tion algorithm (ICALP'99) and a re-interpretation of Regev's quantum reduction between the Bounded Distance Decoding problem and sam- pling short lattice vectors.
Abstract. The decision Learning With Errors problem has proven an extremely flexible foundation for devising provably secure cryptographic primitives. LWE can be expressed in terms of linear algebra over Z/qZ. This modulus q is the subject of study of the present work. When q is prime and small, or when it is exponential and composite with small factors, LWE is known to be at least as hard as standard worst-case problems over euclidean lattices (sometimes using quantum reductions). The Ring Learning With Errors problem is a structured variant of LWE allowing for more compact keys and more efficient primitives. It is known to be at least as hard as standard worst-case problems restricted to so-called ideal lattices, but under even more restrictive arithmetic conditions on q. In this work, we prove that the arithmetic form of the modulus q is irrelevant to the computational hardness of LWE and RLWE. More precisely, we show that these problems are at least as hard as standard worst-case problems on lattices, under the unique condition that q is of polynomial bit-size. This result is most useful for adapting LWE-based cryptographic constructions to the RLWE setting. Among others, this allows us to derive the first Identity-Based Encryption scheme of quasi-optimal performance proven secure under standard worst-case lattice assumptions, in the standard model. Other applications include authentication, functional encryption and traitor tracing.
Lattice-based cryptography is the use of conjectured hard problems on point lattices in Rn as the foundation for secure cryptographic systems. Attractive features of lattice cryptography include apparent resistance to quantum attacks (in contrast with most number-theoretic cryptography), high asymptotic efficiency and parallelism, security under worst-case intractability assumptions, and solutions to long-standing open problems in cryptography. This monograph surveys most of the major developments in lattice cryptography over the past ten years. The main focus is on the foundational short integer solution (SIS) and learning with errors (LWE) problems (and their more efficient ring-based variants), their provable hardness assuming the worst-case intractability of standard lattice problems, and their many cryptographic applications.