We present a lattice-based stateless signature scheme provably secure in the standard model. Our scheme has a constant number of matrices in the public key and a single lattice vector (plus a tag) in the signatures. The best previous lattice-based encryption schemes were the scheme of Ducas and Micciancio (CRYPTO 2014), which required a logarithmic number of matrices in the public key and that of Bohl et. al (J. of Cryptology 2014), which required a logarithmic number of lattice vectors in the signature. Our main technique involves using fully homomorphic computation to compute a degree d polynomial over the tags hidden in the matrices in the public key. In the scheme of Ducas and Micciancio, only functions linear over the tags in the public key matrices were used, which necessitated having d matrices in the public key. As a matter of independent interest, we extend Wichs ’ (eprint 2014) recent construction of homomor-phic trapdoor functions into a primitive we call puncturable homomorphic trapdoor functions (PHTDFs). This primitive abstracts out most of the properties required in many different lattice-based cryptographic constructions. We then show how to combine a PHTDF along with a function satisfying certain properties (to be evaluated homomorphically) to give an eu-scma signature scheme.
We present a lattice-based stateless signature scheme provably secure in the standard model. Our scheme has a constant number of matrices in the public key and a single lattice vector (plus a tag) in the signatures. The best previous lattice-based encryption schemes were the scheme of Ducas and Micciancio (CRYPTO 2014), which required a logarithmic number of matrices in the public key and that of Bohl et. al (J. of Cryptology 2014), which required a logarithmic number of lattice vectors in the signature. Our main technique involves using fully homomorphic computation to compute a degree d polynomial over the tags hidden in the matrices in the public key. In the scheme of Ducas and Micciancio, only functions linear over the tags in the public key matrices were used, which necessitated having d matrices in the public key. As a matter of independent interest, we extend Wichs ’ (eprint 2014) recent construction of homomor-phic trapdoor functions into a primitive we call puncturable homomorphic trapdoor functions (PHTDFs). This primitive abstracts out most of the properties required in many different lattice-based cryptographic constructions. We then show how to combine a PHTDF along with a function satisfying certain properties (to be evaluated homomorphically) to give an eu-scma signature scheme.
Digital signatures and key exchange protocols are the two most important public key cryptographic primitives used in the electronic transmission of data. The goal of key exchange is to preserve the secrecy of the communication, while the goal of digital signatures is to guarantee the authenticity of the exchanged messages. Constructions of digital signature schemes based on classical mathematical assumptions appeared shortly following the invention of public key cryptography in the late 1970s. And just like with key exchange, the most efficient variants are based on number-theoretic problems that are believed to be (sub)-exponentially hard for classical machines, but are solved in polynomial time by Shor’s algorithm on a powerful-enough quantum computer. Also, like for key exchange, the most efficient constructions that we believe to be quantum safe are based on the presumed hardness of lattice problems over polynomial rings. One interesting difference between key exchange and digital signatures is that key exchange appears to inherently require that some mathematical problem be computationally hard. Digital signatures, on the other hand, can be generically constructed from any one-way function [1,2]. So even though they certainly fall into the category of public key primitives based on their usage, their existence requires much weaker assumptions. Additionally, the transformation from a one-way function to a digital signature is not too inefficient. For example, the total parameter size (public key + signature) of the SPHINCS+ scheme [3] is around 40 kB. While these sizes are larger, and signing times considerably longer, than those of signatures based on factoring or discrete log, it is still a usable scheme for many applications. And being only based on symmetric assumptions (e.g. one wayness and collision resistance of cryptographic hash functions), its security is very attractive. In order to be considered an interesting alternative to the above-mentioned signature, a scheme based on a mathematical assumption would need to have significant performance advantages. Schemes based on factoring and the discrete logarithm problem were significantly shorter and faster, and so the generic approach lay dormant for over four decades. The new quantum-safe schemes will need to have similar performance advantages if they are to be used in lieu of this safe approach. Below, we describe two techniques for constructing lattice-based digital signatures with output sizes being just a few kilobytes. On a very high level, lattice-based signature constructions follow the two known approaches for constructing classical signatures. In the first approach, the signer outputs a function f and an image y = f(x) as his public key and keeps x as his secret key. To sign a message μ, he gives a non-interactive zero-knowledge proof that he knows an x satisfying y = f(x), using the message μ to create the ‘challenge’ H(μ) for the proof (where H is a public function that maps μ to something ‘random looking’). If the function f is one way then the verifier should be convinced that the proof could have only been created by the entity who knows x. A classic example of this type of scheme is the Schnorr signature scheme [4] based on the hardness of the discrete logarithm problem. The second approach is to create a function f together with a trapdoor f−1, output f as the public key and keep f−1 as the secret key. A message μ is signed by using the secret trapdoor to create a pre-image x such that f(x) = H(μ). Again, if the function f is one way then only someone in possession of a trapdoor should be able to invert it. An example of such a construction based on a ‘factoring-like’ assumption is the RSA signature scheme [5]. The high-level ideas for lattice-based signatures follow the above blueprints, but the technical details are significantly more involved. The main reason for the complications is the different algebraic structure of the hard one-way function underlying lattice cryptography. While the domain of the function in discrete log and RSA-based one-way functions are groups, the domains in lattice-based signatures are sets that are not closed under any operation—in particular, they are elements in a group that have small norms. This crucial small norm requirement precludes us from using uniformly random masking as in Schnorr signatures or having a trapdoor for a bijective one-way function as in RSA signatures. These barriers have, nevertheless, been overcome and the resulting digital signatures are quite practical. In 2017, the US National Institute of Standards and Technology (NIST) began a ‘competition’ for a quantum-safe key exchange and digital signatures standard. At the time of this writing, this process is in the third round and there are two lattice-based signatures remaining—each following one of the above high-level designs. The CRYSTALS-Dilithium [6] scheme follows the Schnorr framework, but adds a crucial rejection-sampling step to keep the size of the coefficients small. The FALCON scheme [7] utilizes a randomized trapdoor sampling technique that uses a secret trapdoor for f−1 to produce random pre-images from a particular distribution. Because there is no longer a bijection, it is crucial to also have the property that the distribution of the outputted pre-images does not leak information about the trapdoor. Both schemes are relatively fast and their parameters (public key + signature size) are the shortest of all quantum-safe signature schemes. While both schemes are based on lattices, they have rather different characteristics. FALCON has very short parameters (see Table 1), but entails a rather complicated procedure for signature generation. In particular, it uses (an optimized version of) the GPV sampler [8], which requires floating-point arithmetic with approximately 64 bits of precision. Requiring such high precision means that subtle implementation errors may not get detected even with rigorous testing. Dilithium, on the other hand, has larger parameters but a very simple implementation where all the sampling in the signing is done in a power-of-2 range; it is thus much less prone to implementation errors. It is quite possible that in the future both schemes will be used for different applications. Approximate parameter sizes (in bytes) for the CRYSTALS-Dilithium and FALCON digital signature schemes at approximately 128-bit security levels. Approximate parameter sizes (in bytes) for the CRYSTALS-Dilithium and FALCON digital signature schemes at approximately 128-bit security levels. Conflict of interest statement. None declared.
We propose a short signature scheme under the ring-SIS assumption in the standard model. Specifically, by revisiting an existing construction [Ducas and Micciancio, CRYPTO 2014], we demonstrate lattice-based signatures with improved reduction loss. As far as we know, there are no ways to use multiple tags in the signature simulation of security proof in the lattice tag-based signatures. We address the tag-collision possibility in the lattice setting, which improves reduction loss. Our scheme generates tags from messages by constructing a scheme under a mild security condition that is existentially unforgeable against random message attack with auxiliary information. Thus our scheme can reduce the signature size since it does not need to send tags with the signatures. Our scheme has short signature sizes of O(1) and achieves tighter reduction loss than that of Ducas et al.'s scheme. Our proposed scheme has two variants. Our scheme with one property has tighter reduction and the same verification key size of O(log n) as that of Ducas et al.'s scheme, where n is the security parameter. Our scheme with the other property achieves much tighter reduction loss of O(Q/n) and verification key size of O(n), where Q is the number of signing queries.
Although the fully homomorphic signature (FHS) has made great progress, the low efficiency of existing FHS schemes as one main drawback cannot reach the requirements in practical application. It is well known that Number-Theory-Research-Unit (NTRU)-based cryptography is not only considered to be resistant to the quantum computer but also has high efficiency compared with the standard lattice-based cryptographic systems. In this paper, our goal is to construct a simple and highly efficient FHS scheme. To this end, we first devise an NTRU-based homomorphic trapdoor function (HTDF), which is more effective than the one proposed by Gorbunov et al. (STOC 2015). We then convert it into an efficient NTRU-based FHS scheme. Additionally, the HTDF we constructed is not only collision-resistant but is also claw-free, resulting in our FHS scheme is being strongly unforgeable.
Proxy signature schemes safely ensure that an original signer transfers the signing right to a proxy signer, and a variety of these schemes have been developed for secure applications. Constructions of proxy signature schemes vary; recently developed schemes utilize lattices to safeguard against quantum attacks. However, existing lattice-based proxy schemes proved secure in the standard model have many public system parameters, reducing the efficiency of these schemes. To solve this problem, we utilize an admissible hash function with compatible algorithms (AHFwCA) and lattice trapdoors as an initial framework to propose a new lattice-based proxy signature scheme. While maintaining a similar key and signature size as prior schemes, this new proxy signature scheme requires fewer public system parameters. Compared with requiring linear basic matrices as public system parameters found in other standard model schemes, the proposed new scheme depends upon logarithmic basic matrices. Under the standard inhomogeneous short integer solution (ISIS) assumption, we prove that the scheme is existentially unforgeable against chosen message attacks and chosen warrant attacks (EUF-CMA-CWA) in the standard model by using the generalized partitioning proof technique. Thus, this paper partially solves the many parameters problem existed in the standard model lattice-based signature schemes.
Abstract. We provide an alternative method for constructing lattice-based digital signatures which does not use the “hash-and-sign ” methodology of Gentry, Peikert, and Vaikuntanathan (STOC 2008). Our resulting signature scheme is secure, in the random oracle model, based on the worst-case hardness of the Õ(n1.5)-SIVP problem in general lattices. The secret key, public key, and the signature size of our scheme are smaller than in all previous instantiations of the hash-and-sign signature, and our signing algorithm is also quite simple, requiring just a few matrix-vector multiplications and rejection samplings. We then also show that by slightly changing the parameters, one can get even more efficient signatures that are based on the hardness of the Learning With Errors problem. Our construction naturally transfers to the ring setting, where the size of the public and secret keys can be significantly shrunk, which results in the most practical to-date provably secure signature scheme based on lattices. 1
So far, most ring signature schemes rely on hard number theory problems, such as discrete logarithm, bilinear pairings and so on. Unfortunately, the above underlying number theory problems will be solvable in the post quantum era. Lattice-based cryptography is a hotspot of research recently, due to its implementation simplicity and provable security reductions. When the hash-and-sign signature scheme was constructed based on the hardness of worst-case lattice problems, provably secure lattice-based ring signature schemes were finally constructed. However, the hash-and-sign ring signatures were rather inefficient (with megabytes long signatures). In this paper, we propose an alternative method for constructing lattice-based and identity-based ring signature scheme which does not use the hash-and-sign methodology. In the random oracle model, the proposed signature scheme based on the problem in general lattices is unforgeable and holds anonymity. Compared with the previous instantiations of the hash-and-sign ring signature schemes, the lengths of secret key, public key and signatures in the proposed scheme are much shorter. The signing algorithm is quite simple, with matrix-vector multiplications and rejection samplings.