To provide a high-security guarantee to network coding and lower the computing complexity induced by signature scheme, we take full advantage of homomorphic property to build lattice signature schemes and secure network coding algorithms. Firstly, by means of the distance between the message and its signature in a lattice, we propose a Distance-based Secure Network Coding (DSNC) algorithm and stipulate its security to a new hard problem Fixed Length Vector Problem (FLVP), which is harder than Shortest Vector Problem (SVP) on lattices. Secondly, considering the boundary on the distance between the message and its signature, we further propose an efficient Boundary-based Secure Network Coding (BSNC) algorithm to reduce the computing complexity induced by square calculation in DSNC. Simulation results and security analysis show that the proposed signature schemes have stronger unforgeability due to the natural property of lattices than traditional Rivest-Shamir-Adleman (RSA)-based signature scheme. DSNC algorithm is more secure and BSNC algorithm greatly reduces the time cost on computation.
To provide a high-security guarantee to network coding and lower the computing complexity induced by signature scheme, we take full advantage of homomorphic property to build lattice signature schemes and secure network coding algorithms. Firstly, by means of the distance between the message and its signature in a lattice, we propose a Distance-based Secure Network Coding (DSNC) algorithm and stipulate its security to a new hard problem Fixed Length Vector Problem (FLVP), which is harder than Shortest Vector Problem (SVP) on lattices. Secondly, considering the boundary on the distance between the message and its signature, we further propose an efficient Boundary-based Secure Network Coding (BSNC) algorithm to reduce the computing complexity induced by square calculation in DSNC. Simulation results and security analysis show that the proposed signature schemes have stronger unforgeability due to the natural property of lattices than traditional Rivest-Shamir-Adleman (RSA)-based signature scheme. DSNC algorithm is more secure and BSNC algorithm greatly reduces the time cost on computation.
Abstract The property that linearly homomorphic signature (LHS) enables linear computation on signed data makes this primitive commonly used to secure the network coding based application against pollution attacks. The security of LHS wholly depends on the security of secret signing keys; once a secret key is compromised, the application system’ security will be broken. However, as relatively insecure mobile devices are increasingly used in network coding application systems, the key exposure issue is becoming more prevalent. On the other hand, with the quantum technology development, some existing LHS schemes based on some traditional number-theoretic primitives are susceptible to quantum cryptanalysis. Hence, facing the dual threats of the exposure of secret keys in reality and the future effiffifficient quantum computers, we integrate forward security into LHS, and present a forward-secure identity-based LHS scheme (FSIBLHS) based on lattice assumptions, which achieves post-quantum security, and even if the current secret key is compromised, the validity of signatures of previous time periods is still guaranteed. We defifine the fifirst security model on forward security of LHS, and prove that the proposed scheme can resist adaptively chosen identity and datasets attack under lattice assumption. Moreover, compared with previous related works, the performance analysis shows that our scheme enjoys a comparable computation cost and has a lower communication cost, and provides higher security and stronger functionality.
Compared to the classical cryptography, lattice-based cryptography is more secure, flexible and simple, and it is believed to be secure against quantum computers.In this paper, an efficient signature scheme is proposed from the ring learning with errors (R-LWE), which avoids sampling from discrete Gaussians and has the characteristics of the much simpler description etc.Then, the scheme is implemented in C/C++ and makes a comparison with the RSA signature scheme in detail.Additionally, a linearly homomorphic signature scheme without trapdoor is proposed from the R-LWE assumption.The security of the above two schemes are reducible to the worst-case hardness of shortest vectors on ideal lattices.The security analyses indicate the proposed schemes are unforgeable under chosen message attack model, and the efficiency analyses also show that the above schemes are much more efficient than other correlative signature schemes.
We propose a linearly homomorphic signature scheme that authenticates vector subspaces of a given ambient space. Our system has several novel properties not found in previous proposals: • It is the first such scheme that authenticates vectors defined over binary fields; previous proposals could only authenticate vectors with large or growing coefficients. • It is the first such scheme based on the problem of finding short vectors in integer lattices, and thus enjoys the worst-case security guarantees common to lattice-based cryptosystems. Our scheme can be used to authenticate linear transformations of signed data, such as those arising when computing mean and Fourier transform or in networks that use network coding. Our construction gives an example of a cryptographic primitive — homomorphic signatures over F2 — that can be built using lattice methods, but cannot currently be built using bilinear maps or other traditional algebraic methods based on factoring or discrete log type problems. Security of our scheme (in the random oracle model) is based on a new hard problem on lattices, called k-SIS, that reduces to standard average-case and worst-case lattice problems. Our formulation of the k-SIS problem adds to the “toolbox” of lattice-based cryptography and may be useful in constructing other lattice-based cryptosystems. As a second application of the new k-SIS tool, we construct an ordinary signature scheme and prove it k-time unforgeable in the standard model assuming the hardness of the k-SIS problem. Our construction can be viewed as “removing the random oracle” from the signatures of Gentry, Peikert, and Vaikuntanathan at the expense of only allowing a small number of signatures.
This paper delves into lattice-based cryptography as a cornerstone of post-quantum cryptography (PQC), exploring its evolution, key principles, and applications. It examines how lattice-based schemes leverage the inherent hardness of problems such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE) to provide quantum-resistant encryption, digital signatures, and homomorphic encryption. The paper also outlines the historical development of lattice-based cryptography, emphasizing its resilience to quantum attacks in contrast to traditional schemes like RSA and ECC, which are vulnerable to Shor's algorithm. Core concepts, including lattice trapdoors, parameter optimization, and real-world applications in blockchain, cryptocurrencies, and secure communication protocols, are explored in detail. Furthermore, the paper highlights ongoing standardization efforts led by NIST, ethical considerations, practical challenges, and future research directions aimed at improving efficiency, ensuring interoperability, and expanding lattice cryptography’s use in advanced security frameworks. Through comprehensive analysis, this research underscores the importance of lattice-based cryptography as a critical pillar for securing the digital ecosystem against the emerging threat of quantum computing.
We propose a variant version of ring learning with errors (R-LWE) assumption. Under the modified slightly assumption which is reducible to the worst-case problems on ideal lattice, we present a construction of digital signatures. So the scheme is provably secure based on the hardness of lattice problems (such as approximating the length of the shortest vector within a fixed poly(n) factor). Compared with some existing typical lattice-based signature schemes, the construction enjoys many advantages, including simple and efficient key generation algorithm, signing algorithm and verification algorithm to improve the efficiency of the scheme as well as to reduce the overhead. The sizes of the secret and public keys and the signature are almost linear O(n log n) (up to poly-logarithmic factors) in the dimension n of the lattice.
In order to guarantee the security of signature scheme under quantum attack,most of the existing lattice-based signature schemes are based on the difficulty hypothesis under general lattice rather than ideal lattice,and it caused lower efficiency of these schemes.To overcome the problem,a new digital signature scheme based on ideal lattice is proposed.By reducing the problem to the PLWE,it is proved that an attacker can not forge a legitimate signature for a new message even if he gets multiple existing messages and the corresponding signature.Analysis result shows that the signature length and the key length of the scheme are reduced by O(n),compared with the existing digital signature scheme based on general lattice.