Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Symposium on Computers and Communications (ISCC)

A public key algorithm for optical communication based on lattice cryptography

Walid Abdallah, Mohamed Hamdi, Noureddine A. Boudriga

Published Jul 1, 2009• 6 citations
Publisher

Abstract

This paper proposes a security architecture for optical code division multiple access networks based on lattice cryptography. Unlike existing approaches, which have mainly focused on optical components such as phase masks and delay lines, our cryptosystem encompasses a secure code construction process. In fact, given a set of optical orthogonal codewords, we associate a pair of bases (a public basis and a private basis) to every user and we show that the projection of a codeword on the public basis of the receiver and the addition of a random error enhances the confidentiality performance of the code. The proposed public key cryptosystem is based on lattice cryptography. The security of this scheme relies mainly on the complexity of the closest vector-problem in an integer lattice. We found that our technique performs better than the existing approaches in terms of robustness to cryptanalysis. We also study the security of our lattice cryptosystem with regard to the properties of the bases pairs and the error vectors.

DOI
10.1109/iscc.2009.5202365

Related papers

Physical layer cryptography in optical networks: A lattice-based approach

Noureddine A. Boudriga, Walid Abdallah, Mohamed Hamdi

This paper discusses the use of lattice cryptography and chaotic maps to encrypt OCDMA-coded optical flows. Unlike existing approaches, which have mainly focused on optical components such as phase masks and delay lines, our cryptosystem encompasses a secure code construction process. In fact, assuming that a pair of bases (a public basis and a private basis) is associated to every user, the OCDMA code-words are projected on a public basis before being sent so that only the user possessing the corresponding private basis can decrypt the keyword. We show that the projection of a codeword on the public basis of the receiver and the addition of a chaotic error enhances the confidentiality performance of the code. This distribution of this error is generated using a chaotic map. In this work, we study the impact of the parameters of this chaotic map on the robustness of the proposed cryptosystem.

International Conference on Transparent Optical Networks•Jun 1, 2010•4 citations

An all optical configurable and secure OCDMA system implementation using loop based optical delay lines

Walid Abdallah, Mohamed Hamdi, Noureddine A. Boudriga

This paper describes the physical implementation of an all-optical implementation security architecture for optical code division multiple access networks based on lattice cryptography. Our design is mainly based on the use of loop-based optical delay lines and fiber Bragg grating. The encryption process is achieved by delaying a pulse in the optical delay line according to the structure of the codewords. The number of loops is defined by the position of the pulse in codeword. We demonstrate that our implementation is less complex and more flexible than classical approaches such as fixed length optical delay lines. In addition, a performance evaluation using eye diagrams shows that our architecture is more robust against eavesdropping security attacks.

International Conference on Transparent Optical Networks•Jun 1, 2011•3 citations

Orthogonal DPSK/CSK Modulation and Public-Key Cryptography-Based Secure Optical Communication

Bo Dai, Zhensen Gao, Naoya Wada +1 more

In this letter, the information security of an optical code-based communication system employing orthogonal differential phase-shift keying (DPSK)/code-shift keying (CSK) modulation is first investigated using two eaves dropping methods, one-bit delay interference detection and differential detection. The investigation results show that the security vulnerability exists only in the CSK channel while the DPSK channel can thwart these attacks. Therefore, the concept of public-key cryptography is introduced into the system to efficiently use these two orthogonal channels for both public key distribution and encrypted data transmission. The privacy of the transmitted data can be double guaranteed by public-key cryptography scheme logically and optical code-processing techniques physically.

IEEE Photonics Technology Letters•Aug 26, 2013•8 citations

Lattice-Based Cryptosystems

Zhuo Liu, Yongkun Zhao, Lingcai Huang

This chapter introduces lattice-based cryptography, a critical mathematical framework for securing communications in the post-quantum era. While classical public-key cryptosystems rely on the intractability of integer factorization and discrete logarithms, these foundations are fundamentally vulnerable to quantum computing threats, specifically Shor's algorithm. To address this vulnerability, the text explores cryptosystems built upon the algebraic structure of lattices, defined as discrete points in $n$-dimensional Euclidean space generated by linearly independent vectors. The chapter systematically covers the mathematical properties of lattices, associated computational problems, and distance metrics. It provides an in-depth examination of lattice basis reduction algorithms, which are fundamental to both the design and security evaluation of these systems. Furthermore, the chapter details the operational mechanics—including encryption, decryption, and cryptanalysis—of prominent post-quantum architectures, specifically the GGH and NTRU public-key cryptosystems. By transitioning from classical number theory to high-dimensional geometry, this chapter equips readers with the cryptographic primitives necessary to withstand future quantum attacks.

CRC Press eBooks•Sep 18, 2026•0 citations

Learning with Errors and Lattice-Based Post-Quantum Public-Key Cryptography

Jaskari, Mikko

The security of conventional public-key cryptography schemes is based on mathematical problems which are considered to be infeasible to solve by classical computers in a reasonable amount of time. However, the rise of quantum computers might break this security in the future. Post-quantum cryptography (PQC) aims to achieve secure schemes which are resistant against attacks assisted by quantum computers. Public-key cryptography refers to asymmetric cryptography in which encryption is done with a public key and decryption is done with a private key. Characteristic properties of public-key cryptography include the possibility to derive the private key from public information by solving some presumably hard problem. The hardness of the problem constitutes the basis for the system's security. In the post-quantum era, the security must be based on problems that are too hard to solve even for quantum computers. In this thesis, we will study lattice-based cryptography which is believed to serve as a good candidate for secure post-quantum public-key cryptography. Our main attention is in the problem of learning with errors, which is in the base of some lattice-based schemes and is linked to the hard problem of finding short vectors in a lattice.

•Dec 5, 2025•0 citations

Lattice-based PKEs/KEMs

Xianhui Lu, Jiang Zhang

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.

National Science Review•May 20, 2021•1 citation
This paper proposes a security architecture for optical code division multiple access networks based on lattice cryptography. Unlike existing approaches, which have mainly focused on optical components such as phase masks and delay lines, our cryptosystem encompasses a secure code construction process. In fact, given a set of optical orthogonal codewords, we associate a pair of bases (a public basis and a private basis) to every user and we show that the projection of a codeword on the public basis of the receiver and the addition of a random error enhances the confidentiality performance of the code. The proposed public key cryptosystem is based on lattice cryptography. The security of this scheme relies mainly on the complexity of the closest vector-problem in an integer lattice. We found that our technique performs better than the existing approaches in terms of robustness to cryptanalysis. We also study the security of our lattice cryptosystem with regard to the properties of the bases pairs and the error vectors.
This paper discusses the use of lattice cryptography and chaotic maps to encrypt OCDMA-coded optical flows. Unlike existing approaches, which have mainly focused on optical components such as phase masks and delay lines, our cryptosystem encompasses a secure code construction process. In fact, assuming that a pair of bases (a public basis and a private basis) is associated to every user, the OCDMA code-words are projected on a public basis before being sent so that only the user possessing the corresponding private basis can decrypt the keyword. We show that the projection of a codeword on the public basis of the receiver and the addition of a chaotic error enhances the confidentiality performance of the code. This distribution of this error is generated using a chaotic map. In this work, we study the impact of the parameters of this chaotic map on the robustness of the proposed cryptosystem.
This paper describes the physical implementation of an all-optical implementation security architecture for optical code division multiple access networks based on lattice cryptography. Our design is mainly based on the use of loop-based optical delay lines and fiber Bragg grating. The encryption process is achieved by delaying a pulse in the optical delay line according to the structure of the codewords. The number of loops is defined by the position of the pulse in codeword. We demonstrate that our implementation is less complex and more flexible than classical approaches such as fixed length optical delay lines. In addition, a performance evaluation using eye diagrams shows that our architecture is more robust against eavesdropping security attacks.
In this letter, the information security of an optical code-based communication system employing orthogonal differential phase-shift keying (DPSK)/code-shift keying (CSK) modulation is first investigated using two eaves dropping methods, one-bit delay interference detection and differential detection. The investigation results show that the security vulnerability exists only in the CSK channel while the DPSK channel can thwart these attacks. Therefore, the concept of public-key cryptography is introduced into the system to efficiently use these two orthogonal channels for both public key distribution and encrypted data transmission. The privacy of the transmitted data can be double guaranteed by public-key cryptography scheme logically and optical code-processing techniques physically.
This chapter introduces lattice-based cryptography, a critical mathematical framework for securing communications in the post-quantum era. While classical public-key cryptosystems rely on the intractability of integer factorization and discrete logarithms, these foundations are fundamentally vulnerable to quantum computing threats, specifically Shor's algorithm. To address this vulnerability, the text explores cryptosystems built upon the algebraic structure of lattices, defined as discrete points in nnn-dimensional Euclidean space generated by linearly independent vectors. The chapter systematically covers the mathematical properties of lattices, associated computational problems, and distance metrics. It provides an in-depth examination of lattice basis reduction algorithms, which are fundamental to both the design and security evaluation of these systems. Furthermore, the chapter details the operational mechanics—including encryption, decryption, and cryptanalysis—of prominent post-quantum architectures, specifically the GGH and NTRU public-key cryptosystems. By transitioning from classical number theory to high-dimensional geometry, this chapter equips readers with the cryptographic primitives necessary to withstand future quantum attacks.
The security of conventional public-key cryptography schemes is based on mathematical problems which are considered to be infeasible to solve by classical computers in a reasonable amount of time. However, the rise of quantum computers might break this security in the future. Post-quantum cryptography (PQC) aims to achieve secure schemes which are resistant against attacks assisted by quantum computers. Public-key cryptography refers to asymmetric cryptography in which encryption is done with a public key and decryption is done with a private key. Characteristic properties of public-key cryptography include the possibility to derive the private key from public information by solving some presumably hard problem. The hardness of the problem constitutes the basis for the system's security. In the post-quantum era, the security must be based on problems that are too hard to solve even for quantum computers. In this thesis, we will study lattice-based cryptography which is believed to serve as a good candidate for secure post-quantum public-key cryptography. Our main attention is in the problem of learning with errors, which is in the base of some lattice-based schemes and is linked to the hard problem of finding short vectors in a lattice.
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.