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Science China Information Sciences

Lattice-based group encryptions with only one trapdoor

Jing Pan, Jiang Zhang, Fangguo Zhang, Xiaofeng Chen, Willy Susilo

Published Mar 16, 2022• 3 citations
Publisher
DOI
10.1007/s11432-020-3226-6

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Group communication enables Internet of Things (IoT) devices to communicate in an efficient and fast manner. In most instances, a group message needs to be encrypted using a cryptographic key that only devices in the group know. In this paper, we address the problem of establishing such a key using a lattice-based one-way function, which can easily be inverted using a suitably designed lattice trapdoor. Using the notion of a bad/good basis, we present a new method of coupling multiple private keys into a single public key, which is then used for encrypting a group message. The protocol has the apparent advantage of having a conjectured resistance against potential quantum-computer-based attacks. All functions-key establishment, session key update, node addition, encryption, and decryption-are effected in constant time, using simple linear-algebra operations, making the protocol suitable for resource-constrained IoT networks. We show how a cryptographic session group key can be constructed on the fly by a user with legitimate credentials, making node-capture-type attacks impractical. The protocol also incorporates a mechanism for node addition and session-key generation in a forward- and backward-secrecy-preserving manner.

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Losing information

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In the first part of this work, Chapter 3 we present new and general constructions of lossy encryption schemes. The Selective Opening problem is as follows: suppose an adversary receives n commitments (or encryptions) of (possibly) correlated messages, and now the adversary can choose n/2 of the messages, and receive de-commitments (or decryptions and the randomness used to encrypt them). Do the unopened commitments (encryptions) remain secure? A protocol achieving this type of security is called secure against a selective opening adversary (SOA). This question arises naturally in the context of Byzantine Agreement and Secure Multiparty Computation, where an active adversary is able to eavesdrop on all the wires, and then choose a subset of players to corrupt. Unfortunately, the traditional definitions of security (IND-CPA, IND-CCA) do not guarantee security in this setting. In this paper: · We formally define re-randomizable encryption and show that every re-randomizable encryption scheme gives rise to efficient encryptions secure against a selective opening adversary. (Very informally, an encryption is re-randomizable, if given any ciphertext, there is an efficient way to map it to an almost uniform re-encryption of the same underlying message). · We show that statistically-hiding 2-round Oblivious Transfer (OT) implies Lossy Encryption and so do smooth hash proof systems, as defined by Cramer-Shoup. Combining this with known results immediately shows that Private Information Retrieval and homomorphic encryption both imply Lossy Encryption, and thus Selective Opening Secure Public Key Encryption. In the second part of this work, Chapter 4, we focus on constructing lossy trapdoor functions. Lossy trapdoor functions (LTDFs) were introduced by Peikert and Waters in STOC '08. Since their introduction, lossy trapdoor functions have found many uses in cryptography. In the work of Peikert and Waters, lossy trapdoor functions were used to give an efficient construction of a chosen-ciphertext secure (IND-CCA2) cryptosystem. Lossy trapdoor functions were then shown to imply deterministic encryption by Boldyreva, Fehr and O'Neill in CRYPTO '08. In TCC '09, Rosen and Segev showed that lossy trapdoor functions are correlated product secure, meaning that they remain one-way even when evaluated on correlated inputs. In this section we extend the notion of smooth hash proof systems as defined by Cramer and Shoup in Eurocrypt '02, to include an additional homomorphic property. We call this primitive smooth homomorphic hash proof systems . We show that smooth homomorphic projective hash proof systems include all Diverse Group Systems, as defined by Cramer and Shoup. Using this definition, we show that · Smooth homomorphic hash proof systems imply LTDFs. · Diverse group systems as defined in [CS02] imply LTDFs. These are the first known generic constructions of LTDFs. · Applying our generic construction the specific constructions of smooth hash proof systems given by Cramer and Shoup, we obtain the first construction of fully lossy trapdoor functions from the quadratic residuosity (QR) assumption. We also obtain a novel construction of LTDFs from Paillier's decisional composite residuosity (DCR) assumption. · Applying the black-box separation results of Rosen and Segev, we show that there is a black-box separation between smooth homomorphic hash proof systems and one-way trapdoor permutations. In the third part of this work, Chapter 5, we examine in what situations we can construct lossy trapdoor functions from the seemingly weaker primitive lossy encryption. Injective one-way trapdoor functions are one of the most fundamental cryptographic primitives. In this section we give a novel construction of injective trapdoor functions based on oblivious transfer for long strings. Our main result is to show that any 2-message statistically sender-private semi-honest oblivious transfer (OT) for strings longer than the sender randomness implies the existence of injective one-way trapdoor functions. This is perhaps surprising given the black box separation of injective one-way trapdoor functions from many common cryptographic protocols, e.g. IND-CCA encryption. As a tool for creating injective one-way trapdoor functions, we define a new notion of security for a public key encryption scheme called Randomness Dependent Message (RDM) security, and use it as a stepping stone for creating injective one-way trapdoor functions. (Abstract shortened by UMI.)

•Jan 1, 2010•1 citation

An Analysis of Lattice-Based Cryptography with Existing Protocols

Atul Chaturvedi, Anand Ballabh Joshi, Sonam Yadav

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Group communication enables Internet of Things (IoT) devices to communicate in an efficient and fast manner. In most instances, a group message needs to be encrypted using a cryptographic key that only devices in the group know. In this paper, we address the problem of establishing such a key using a lattice-based one-way function, which can easily be inverted using a suitably designed lattice trapdoor. Using the notion of a bad/good basis, we present a new method of coupling multiple private keys into a single public key, which is then used for encrypting a group message. The protocol has the apparent advantage of having a conjectured resistance against potential quantum-computer-based attacks. All functions-key establishment, session key update, node addition, encryption, and decryption-are effected in constant time, using simple linear-algebra operations, making the protocol suitable for resource-constrained IoT networks. We show how a cryptographic session group key can be constructed on the fly by a user with legitimate credentials, making node-capture-type attacks impractical. The protocol also incorporates a mechanism for node addition and session-key generation in a forward- and backward-secrecy-preserving manner.
Post-quantum group key exchange protocols have become an increasingly critical area of research due to the potential threat quantum computers pose to classical cryptographic schemes. Traditional group key exchange protocols, often based on Diffie-Hellman and other number-theoretic assumptions, are vulnerable to quantum attacks. To address this challenge, we propose a novel group key exchange protocol that leverages lattice-based cryptography. Specifically, our approach utilizes lattice trapdoors and a re-encryption mechanism based on the Ring Learning With Errors (RLWE) problem. This design not only achieves post-quantum security but also enhances efficiency and flexibility in dynamic group settings. A common challenge in most protocols is the computational cost of deriving new keys as group members join and leave. Our approach effectively reduces this cost, making key updates more efficient and practical. Our protocol is proven secure under standard lattice assumptions, providing a promising solution for quantum-resistant group key exchange.
In the first part of this work, Chapter 3 we present new and general constructions of lossy encryption schemes. The Selective Opening problem is as follows: suppose an adversary receives n commitments (or encryptions) of (possibly) correlated messages, and now the adversary can choose n/2 of the messages, and receive de-commitments (or decryptions and the randomness used to encrypt them). Do the unopened commitments (encryptions) remain secure? A protocol achieving this type of security is called secure against a selective opening adversary (SOA). This question arises naturally in the context of Byzantine Agreement and Secure Multiparty Computation, where an active adversary is able to eavesdrop on all the wires, and then choose a subset of players to corrupt. Unfortunately, the traditional definitions of security (IND-CPA, IND-CCA) do not guarantee security in this setting. In this paper: · We formally define re-randomizable encryption and show that every re-randomizable encryption scheme gives rise to efficient encryptions secure against a selective opening adversary. (Very informally, an encryption is re-randomizable, if given any ciphertext, there is an efficient way to map it to an almost uniform re-encryption of the same underlying message). · We show that statistically-hiding 2-round Oblivious Transfer (OT) implies Lossy Encryption and so do smooth hash proof systems, as defined by Cramer-Shoup. Combining this with known results immediately shows that Private Information Retrieval and homomorphic encryption both imply Lossy Encryption, and thus Selective Opening Secure Public Key Encryption. In the second part of this work, Chapter 4, we focus on constructing lossy trapdoor functions. Lossy trapdoor functions (LTDFs) were introduced by Peikert and Waters in STOC '08. Since their introduction, lossy trapdoor functions have found many uses in cryptography. In the work of Peikert and Waters, lossy trapdoor functions were used to give an efficient construction of a chosen-ciphertext secure (IND-CCA2) cryptosystem. Lossy trapdoor functions were then shown to imply deterministic encryption by Boldyreva, Fehr and O'Neill in CRYPTO '08. In TCC '09, Rosen and Segev showed that lossy trapdoor functions are correlated product secure, meaning that they remain one-way even when evaluated on correlated inputs. In this section we extend the notion of smooth hash proof systems as defined by Cramer and Shoup in Eurocrypt '02, to include an additional homomorphic property. We call this primitive smooth homomorphic hash proof systems . We show that smooth homomorphic projective hash proof systems include all Diverse Group Systems, as defined by Cramer and Shoup. Using this definition, we show that · Smooth homomorphic hash proof systems imply LTDFs. · Diverse group systems as defined in [CS02] imply LTDFs. These are the first known generic constructions of LTDFs. · Applying our generic construction the specific constructions of smooth hash proof systems given by Cramer and Shoup, we obtain the first construction of fully lossy trapdoor functions from the quadratic residuosity (QR) assumption. We also obtain a novel construction of LTDFs from Paillier's decisional composite residuosity (DCR) assumption. · Applying the black-box separation results of Rosen and Segev, we show that there is a black-box separation between smooth homomorphic hash proof systems and one-way trapdoor permutations. In the third part of this work, Chapter 5, we examine in what situations we can construct lossy trapdoor functions from the seemingly weaker primitive lossy encryption. Injective one-way trapdoor functions are one of the most fundamental cryptographic primitives. In this section we give a novel construction of injective trapdoor functions based on oblivious transfer for long strings. Our main result is to show that any 2-message statistically sender-private semi-honest oblivious transfer (OT) for strings longer than the sender randomness implies the existence of injective one-way trapdoor functions. This is perhaps surprising given the black box separation of injective one-way trapdoor functions from many common cryptographic protocols, e.g. IND-CCA encryption. As a tool for creating injective one-way trapdoor functions, we define a new notion of security for a public key encryption scheme called Randomness Dependent Message (RDM) security, and use it as a stepping stone for creating injective one-way trapdoor functions. (Abstract shortened by UMI.)