The NTRU lattice is a promising candidate to construct practical cryptosystems, in particular key encapsulation mechanism (KEM), resistant to quantum computing attacks. Nevertheless, there are still some inherent obstacles to NTRU-based KEM schemes in having integrated performance, taking security, bandwidth, error probability, and computational efficiency \emph{as a whole}, that is as good as and even better than their \{R,M\}LWE-based counterparts. In this work, we solve this problem by presenting a new family of NTRU-based KEM schemes, referred to as CTRU and CNTR. By bridging low-dimensional lattice codes and high-dimensional NTRU-lattice-based cryptography with careful design and analysis, to the best of our knowledge CTRU and CNTR are the first NTRU-based KEM schemes with scalable ciphertext compression via only one \emph{single} ciphertext polynomial, and are the first that could outperform \{R,M\}LWE-based KEM schemes in integrated performance. For instance, compared to Kyber that is currently the only standardized KEM by NIST, on the recommended parameter set CNTR-768 has about $12\%$ smaller ciphertext size while encapsulating 384-bit keys compared to the fixed 256-bit key size of Kyber, security strengthened by $(8,7)$ bits for classical and quantum security respectively, and significantly lower error probability ($2^{-230}$ of CNTR-768 vs. $2^{-164}$ of Kyber-768). In comparison with the state-of-the-art AVX2 implementation of Kyber-768, CNTR-768 is faster by 1.9X in KeyGen, 2.6X in Encaps, and 1.2X in Decaps, respectively. When compared to the NIST Round 3 finalist NTRU-HRSS, our CNTR-768 has about $15\%$ smaller ciphertext size, and the security is strengthened by $(55,49)$ bits for classical and quantum security respectively. As for the AVX2 implementation, CNTR-768 is faster than NTRU-HRSS by 19X in KeyGen, 2.3X in Encaps, and 1.6X in Decaps, respectively.
The NTRU lattice is a promising candidate to construct practical cryptosystems, in particular key encapsulation mechanism (KEM), resistant to quantum computing attacks. Nevertheless, there are still some inherent obstacles to NTRU-based KEM schemes in having integrated performance, taking security, bandwidth, error probability, and computational efficiency \emph{as a whole}, that is as good as and even better than their {R,M}LWE-based counterparts. In this work, we solve this problem by presenting a new family of NTRU-based KEM schemes, referred to as CTRU and CNTR. By bridging low-dimensional lattice codes and high-dimensional NTRU-lattice-based cryptography with careful design and analysis, to the best of our knowledge CTRU and CNTR are the first NTRU-based KEM schemes with scalable ciphertext compression via only one \emph{single} ciphertext polynomial, and are the first that could outperform {R,M}LWE-based KEM schemes in integrated performance. For instance, compared to Kyber that is currently the only standardized KEM by NIST, on the recommended parameter set CNTR-768 has about 12% smaller ciphertext size while encapsulating 384-bit keys compared to the fixed 256-bit key size of Kyber, security strengthened by (8,7) bits for classical and quantum security respectively, and significantly lower error probability (2−230 of CNTR-768 vs. 2−164 of Kyber-768). In comparison with the state-of-the-art AVX2 implementation of Kyber-768, CNTR-768 is faster by 1.9X in KeyGen, 2.6X in Encaps, and 1.2X in Decaps, respectively. When compared to the NIST Round 3 finalist NTRU-HRSS, our CNTR-768 has about 15% smaller ciphertext size, and the security is strengthened by (55,49) bits for classical and quantum security respectively. As for the AVX2 implementation, CNTR-768 is faster than NTRU-HRSS by 19X in KeyGen, 2.3X in Encaps, and 1.6X in Decaps, respectively.
We present BAT – an IND-CCA secure key encapsulation mechanism (KEM) that is based on NTRU but follows an encryption/decryption paradigm distinct from classical NTRU KEMs. It demonstrates a new approach of decrypting NTRU ciphertext since its introduction 25 years ago. Instead of introducing an artificial masking parameter p to decrypt the ciphertext, we use 2 linear equations in 2 unknowns to recover the message and the error. The encryption process is therefore close to the GGH scheme. However, since the secret key is now a short basis (not a vector), we need to modify the decryption algorithm and we present a new NTRU decoder. Thanks to the improved decoder, our scheme works with a smaller modulus and yields shorter ciphertexts, smaller than RSA-4096 for 128-bit classical security with comparable public-key size and much faster than RSA or even ECC. Meanwhile, the encryption and decryption are still simple and fast in spite of the complicated key generation. Overall, our KEM has more compact parameters than all current lattice-based schemes and a practical efficiency. Moreover, due to the similar key pair structure, BAT can be of special interest in some applications using Falcon signature that is also the most compact signature in the round 3 of the NIST post-quantum cryptography standardization. However, different from Falcon, our KEM does not rely on floating-point arithmetic and can be fully implemented over the integers.
In this paper, we present CTRU-Light, an IND-CCA-secure key encapsulation mechanism (KEM) derived from NTRU and RLWE (and RLWR in variant) assumptions over power-of-two cyclotomic rings. Our CTRU-Light employs a compact NTT-compatible modulus q=641 while maintaining minimal public key and ciphertext dimensions with negligible error probability. Specifically, the design yields public key and ciphertext sizes of 1206 bytes under an error probability bound of ≤2−110. When benchmarked against Kyber (NIST’s sole standardized KEM), CTRU-Light demonstrates 23.0–30.0% lower bandwidth consumption, accelerates key generation by at least 6.0%, and achieves over 1.3× speed enhancement in both encapsulation and decapsulation procedures.
In post-quantum cryptography, recently many interesting works were done to improve the efficiency of Public Key Encryption (PKE) and Key Encapsulation Management (KEM) schemes based on NTRU (ANTS '98) lattices by reducing the size of the ciphertext (ct) and the public key (pk), and also by improving the speed of the associated algorithms: BAT (TCHES, 2022), NTTRU (TCHES, 2019) and it's generalization (called here NTRU-ABC) by Duman et al (PKC, 2023), NEV (ASIACRYPT 2023), NTRU+ (IEEE, 2023), CTRU and CNTR (Science Direct 2023). In Module-Falcon signature paper (ASIA CCS 2020), a public key encryption based on MNTRU was proposed but its efficiency was not better than existing schemes as said by their authors. In this work, we propose EagleTru, a new family of MNTRU and NTRU based PKEs and KEMs. The public key is H = D(pG) -1 mod ± q ∈ R l×l q where gcd(p, q) = 1, G is a square matrix invertible in R l×l q
The invention of public-key cryptography (PKC) by Diffie and Hellman in 1976 is one of two milestones marking the beginning of modern cryptography. The security of a PKC system requires that it should be infeasible to compute the private key from a given public key, which in turn is typically guaranteed by the difficulty in solving some cryptographic-friendly mathematical problems. Among those problems, the integer factorization and discrete logarithm problems play pivotal roles in the development of public-key cryptography. In particular, the assumption that there is no polynomial time (classical) algorithm that solves the above two problems constitutes the basis for the security of almost all currently used public-key cryptosystems, such as RSA and ElGamal. However, Shor [1] found an efficient quantum solving algorithm for the integer factorization and discrete logarithm problems in 1994, which would destroy the security basis for most real deployed PKC systems if large-scale quantum computers become available. The rapid development of quantum technology in recent years suggests that we are getting closer to the quantum crisis of current PKC systems. As a response to the quantum crisis, the research community has made many efforts to post-quantum cryptography (PQC) that are believed to resist quantum computer attacks. Lattice-based cryptography is one of the main directions in this area, and has become the most promising PQC candidate for standardization. The research of lattice-based cryptography dates back to the seminal work of Ajtai [2], which first based the security of cryptographic primitives on the difficulty of solving some lattice problems such as the approximate shortest vector problem and the closest vector problem. After more than two decades of development, lattice-based cryptography has gained substantial progress. In theory, we have witnessed the construction of many powerful cryptographic primitives that were not known before. In practice, the public key size of lattice-based cryptosystems has been significantly reduced from several gigabytes to a few kilobytes. For current applications and standardization, key exchanges, public-key encryptions (PKEs) and signatures are the most desired lattice-based cryptosystems, where the first two are often used to ensure information secrecy, while the last provides information authentication. Note that public-key encryptions are equivalent to two-round key exchanges, also known as the key encapsulation mechanism (KEM). In the following, we focus on lattice-based PKEs/KEMs. The design principle of lattice-based PKEs/KEMs follows two approaches: a trapdoor one-way function and an approximate commutative one-way function. In the first approach, a trapdoor one-way function f and its trapdoor f−1 are generated as the public key and private key. A plaintext m is encrypted as c = f(m), and the ciphertext is decrypted as m = f−1(c). The NTRU scheme and its variants follow such an approach. In the second approach, an approximate commutative function fs and a random input a are generated, the public key is (a, b = fs(a)) and the private key is s. A plaintext m is encrypted as c1 = gr(a), c2 = gr(b) + E(m), where r is a random element, g is an approximate commutative one-way function and E is the encoding of an error-correction code. The ciphertext (c1, c2) is decrypted as m = D(c2 − fs(c1)), where D is the decoding of an error-correction code. The correctness of decryption is guaranteed by the approximate commutative property (gr(b) = gr(fs(a)) ≈ fs(gr(a)) = fs(c1)) and the error-correction code. Learning with errors (LWE) problem-based PKE/KEM schemes and their variants follow such an approach. The standardization of PQC has received substantial support from national funding agencies, such as the European Union projects PQCrypto and SAFEcrypto. The USA National Institute of Standard and Technology (NIST) started the post-quantum standardization project in 2012, organized a worldwide ‘competition’ in 2016 and plans to deliver draft standards before 2024. After two rounds of evaluation, seven schemes were accepted as the round-three candidates on 22 July 2020, of which five are lattice-based schemes. In 2017, the International organization for standardization started a study project of post-quantum cryptography named SD8. Most LWE-based PKE schemes such as Kyber and LAC follow the Lindner-Peikert framework [3] up to the choices of ring structure, noise distribution and parameters. As the size of noise in lattice-based PKEs/EKMs is typically larger than the lower bits of the ciphertext, many schemes often artificially drop a few lower bits to optimize the practical performance without significantly increasing the final decryption noise. Kyber [4] is one of the most promising PKE/KEM candidates in the NIST PQC standardization, whose security is provably based on the difficulty of so-called module LWE (MLWE) problems. The use of MLWE allowed Kyber to obtain additional flexibility and security advantages, while keeping the same efficiency as those based on the ring-LWE problems. The strategy of dropping some lower bits is used to compress both the public-key and ciphertext sizes. For a targeted 128-bit quantum security, the public-key and ciphertext sizes of Kyber is around a single kilobyte. LAC [5] uses several techniques to reduce the modulus as small as possible: it uses a byte-level modulus, embeds powerful error-correcting codes to maximize the noise-modulus rate, provides a different style of modulus to avoid potential attacks for special polynomial ring structures and optimizes the polynomial multiplication algorithm of the bit-level secret and noise. Based on these techniques, LAC reduces the public key to 544–1056 bytes and the ciphertext to 664–1464 bytes. Meanwhile, it achieves similar computational performance compared with NTT technique schemes. By observing the asymmetries in lattice-based cryptosystems, Zhang and Yu et al. [6] proposed asymmetric MLWE and asymmetric MSIS problems, which yield further size-optimized KEMs and signature schemes to those from standard counterparts. Compared to the seven NIST round-three candidates at Category III security, their KEM scheme has parameters (public key + ciphertext size) smaller than Classic McEliece, CRYSTALS-Kyber, NTRU and Saber; their signature scheme has parameters (public key + signature size) smaller than CRYSTALS-Dilithium and Rainbow, but slightly larger than FALCON, where CRYSTALS-Kyber, NTRU, Saber, CRYSTALS-Dilithium and FALCON are lattice-based schemes. One important application of lattice-based PKEs/KEMs is to build authenticated key exchange (AKE), which could provide additional authentication to the established session key. There are several generic ways to construct AKE from lattice-based PKEs/KEMs. In the PKE setting, Xue and Lu et al. [7] proposed a more compact KEM-based framework via a new primitive named ‘2-key KEM’, which shares some item between different ciphertexts. In the only password setting, Zhang and Yu [8] presented a two-round PAKE framework from the splittable-PKEs that can be instantiated from lattices. The only known AKE that does not rely on lattice-based PKEs/KEMs was given in Ref. [9], which has two-round messages and provides implicit authentication. This work was supported by the National Natural Science Foundation of China (6197239 to X.L.; 62022018 and 61932019 to J.Z.) and the National Key Research and Development Program of China (2018YFB0804105 to J.Z.). Conflict of interest statement. None declared.
In the last decade, there has been significant progress in the development of quantum computers, with massive investments by major tech companies. It is assumed that once large-scale quantum computers are built, many commonly used public key cryptosystems will no longer be secure. To prevent quantum attacks, researchers are already working on the design of post-quantum cryptographic protocols, and the US National Institute of Standards and Technology (NIST) is holding an international competition to select new cryptographic standards.Lattice-based cryptographic constructions are promising candidates because they offer strong theoretical security guarantees and can be implemented efficiently. One of the most widely used cryptographic primitives based on lattices is the Learning With Errors (LWE) problem introduced by Regev. Later works have proposed structured variants of LWE such as Ring-LWE and Module-LWE, which allow for a more compact representation.In this thesis, we consider two lattice-based NIST candidates for Key Encapsulation Mechanisms (KEMs) and propose new error correction and reconciliation techniques in order to improve their efficiency, their security, as well as their reliability.Unlike some previous works on error correction for lattice-based protocols, we provide rigorous error probability bounds.We first consider FrodoKEM, a lattice-based cryptosystem based on LWE, and introduce a modified error correction mechanism to improve its performance. Our encoder maps the secret key block-wise into a scaled version of the 8-dimensional Gosset lattice E_8. We propose three sets of parameters for our modified implementation. The first implementation outperforms FrodoKEM in terms of plausible security; the second allows to reduce the bandwidth by halving the modulus, and the third allows to increase key sizes.The second KEM we are considering is KyberKEM, which is based on Module-LWE. We propose a reconciliation technique using the lattice E_8, and show that our scheme can outperform KyberKEM in terms of security with comparable error probability and similar bandwidth requirements. We also investigate the use of higher dimensional lattices for reconciliation.
The advent of quantum computing poses a significant existential threat to current public-key cryptographic standards, particularly RSA and Elliptic Curve Cryptography (ECC), which underpin global digital security. Shor's algorithm, if implemented on a sufficiently powerful quantum computer, can efficiently break these foundational schemes, rendering vast amounts of encrypted data vulnerable. This paper investigates lattice-based cryptography as a robust and promising candidate for post-quantum security. We provide a comprehensive analysis of its theoretical underpinnings, focusing on hard problems like the Shortest Vector Problem (SVP) and Learning With Errors (LWE), which are believed to be intractable even for quantum adversaries. Through a detailed literature review and methodological exposition, we examine the construction principles of prominent lattice-based Key Encapsulation Mechanisms (KEMs) and digital signature schemes, such as Kyber and Dilithium, highlighting their inherent security properties and performance characteristics. Our results demonstrate that lattice-based schemes offer competitive efficiency, compact key sizes, and strong security guarantees against both classical and quantum attacks, positioning them as essential building blocks for the post-RSA cryptographic landscape. The discussion elaborates on the practical implications, standardization efforts by NIST, and the challenges associated with the transition to quantum-resilient cryptographic infrastructure. We conclude that lattice-based cryptography is a cornerstone of quantum resilience, critical for securing future digital communications and data integrity.