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Learning With Errors
In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by introducing noise to it. In more technical terms, it refers to the computational problem of inferring a linear n-ary function f over a finite Ring (mathematics), ring from given samples y_i = f(\mathbf_i) some of which may be erroneous. The LWE problem is conjectured to be hard to solve, and thus to be useful in cryptography. More precisely, the LWE problem is defined as follows. Let \mathbb_q denote the ring of integers Modular arithmetic, modulo q and let \mathbb_q^n denote the set of n-Vector (mathematics and physics), vectors over \mathbb_q . There exists a certain unknown linear function f:\mathbb_q^n \rightarrow \mathbb_q, and the input to the LWE problem is a sample of pairs (\mathbf,y), wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chinese Remainder Theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The theorem is sometimes called Sunzi's theorem. Both names of the theorem refer to its earliest known statement that appeared in '' Sunzi Suanjing'', a Chinese manuscript written during the 3rd to 5th century CE. This first statement was restricted to the following example: If one knows that the remainder of ''n'' divided by 3 is 2, the remainder of ''n'' divided by 5 is 3, and the remainder of ''n'' divided by 7 is 2, then with no other information, one can determine the remainder of ''n'' divided by 105 (the product of 3, 5, and 7) without knowing the value of ''n''. In this example, the remainder is 23. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kyber
Kyber is a key encapsulation mechanism (KEM) designed to be resistant to cryptanalysis, cryptanalytic attacks with future powerful quantum computing, quantum computers. It is used to establish a shared secret between two communicating parties without an (IND-CCA2) attacker in the transmission system being able to decrypt it. This public-key cryptography, asymmetric cryptosystem uses a variant of the learning with errors lattice problem as its basic trapdoor function. It won the NIST Post-Quantum Cryptography Standardization, NIST competition for the first post-quantum cryptography (PQ) standard. NIST calls its standard, numbered FIPS 203, Module-Lattice-Based Key-Encapsulation Mechanism (ML-KEM). Properties The system is based on the module learning with errors (M-LWE) problem, in conjunction with cyclotomic field, cyclotomic Ring (mathematics), rings. Recently, there has also been a tight formal mathematical security reduction (mathematics), reduction of the ring-LWE problem to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Integer Solution Problem
Short integer solution (SIS) and ring-SIS problems are two ''average''-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Miklós AjtaiAjtai, Miklós. enerating hard instances of lattice problems.Proceedings of the twenty-eighth annual ACM symposium on Theory of computing. ACM, 1996. who presented a family of one-way functions based on SIS problem. He showed that it is secure in an average case if the shortest vector problem \mathrm_\gamma (where \gamma = n^c for some constant c>0) is hard in a worst-case scenario. Average case problems are the problems that are hard to be solved for some randomly selected instances. For cryptography applications, worst case complexity is not sufficient, and we need to guarantee cryptographic construction are hard based on average case complexity. Lattices A ''full rank lattice'' \mathfrak \subset \R^n is a set of integer linear combinations of n linearly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Learning With Errors Key Exchange
In cryptography, a public key exchange algorithm is a cryptographic algorithm which allows two parties to create and share a secret key, which they can use to encrypt messages between themselves. The ring learning with errors key exchange (RLWE-KEX) is one of a new class of public key exchange algorithms that are designed to be secure against an adversary that possesses a quantum computer. This is important because some public key algorithms in use today will be easily broken by a quantum computer if such computers are implemented. RLWE-KEX is one of a set of post-quantum cryptographic algorithms which are based on the difficulty of solving certain mathematical problems involving lattices. Unlike older lattice based cryptographic algorithms, the RLWE-KEX is provably reducible to a known hard problem in lattices. Background Since the 1980s the security of cryptographic key exchanges and digital signatures over the Internet has been primarily based on a small number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice-based Cryptography
Lattice-based cryptography is the generic term for constructions of cryptographic primitives that involve lattices, either in the construction itself or in the security proof. Lattice-based constructions support important standards of post-quantum cryptography. Unlike more widely used and known public-key schemes such as the RSA, Diffie-Hellman or elliptic-curve cryptosystems — which could, theoretically, be defeated using Shor's algorithm on a quantum computer — some lattice-based constructions appear to be resistant to attack by both classical and quantum computers. Furthermore, many lattice-based constructions are considered to be secure under the assumption that certain well-studied computational lattice problems cannot be solved efficiently. In 2024 NIST announced the Module-Lattice-Based Digital Signature Standard for post-quantum cryptography. History In 1996, Miklós Ajtai introduced the first lattice-based cryptographic construction whose security could be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Learning With Errors
In post-quantum cryptography, ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms, such as NewHope, designed to protect against cryptanalysis by quantum computers and also to provide the basis for homomorphic encryption. Public-key cryptography relies on construction of mathematical problems that are believed to be hard to solve if no further information is available, but are easy to solve if some information used in the problem construction is known. Some problems of this sort that are currently used in cryptography are at risk of attack if sufficiently large quantum computers can ever be built, so resistant problems are sought. RLWE is more properly called ''learning with errors over rings'' and is simply the larger learning with errors (LWE) problem specialized to polynomial rings over finite fields. Because of the presumed difficulty of solving the RLWE problem even on a quantum computer, RLWE based c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Post-quantum Cryptography
Post-quantum cryptography (PQC), sometimes referred to as quantum-proof, quantum-safe, or quantum-resistant, is the development of cryptographic algorithms (usually public-key algorithms) that are currently thought to be secure against a cryptanalytic attack by a quantum computer. Most widely-used public-key algorithms rely on the difficulty of one of three mathematical problems: the integer factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem. All of these problems could be easily solved on a sufficiently powerful quantum computer running Shor's algorithm or possibly alternatives. As of 2024, quantum computers lack the processing power to break widely used cryptographic algorithms; however, because of the length of time required for migration to quantum-safe cryptography, cryptographers are already designing new algorithms to prepare for Y2Q or Q-Day, the day when current algorithms will be vulnerable to quantum computin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feige–Fiat–Shamir Identification Scheme
In cryptography, the Feige–Fiat–Shamir identification scheme is a type of parallel zero-knowledge proof developed by Uriel Feige, Amos Fiat, and Adi Shamir in 1988. Like all zero-knowledge proofs, it allows one party, the Prover, to prove to another party, the Verifier, that they possess secret information without revealing to Verifier what that secret information is. The Feige–Fiat–Shamir identification scheme, however, uses modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ... and a parallel verification process that limits the number of communications between Prover and Verifier. Setup Following a common convention, call the prover Peggy and the verifier Victor. Choose two large prime integers ''p'' and ''q'' and compute the product ''n = pq''. Cre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chosen-ciphertext Attack
A chosen-ciphertext attack (CCA) is an attack model for cryptanalysis where the cryptanalyst can gather information by obtaining the decryptions of chosen ciphertexts. From these pieces of information the adversary can attempt to recover the secret key used for decryption. For formal definitions of security against chosen-ciphertext attacks, see for example: Michael Luby and Mihir Bellare et al. Introduction A number of otherwise secure schemes can be defeated under chosen-ciphertext attack. For example, the El Gamal cryptosystem is semantically secure under chosen-plaintext attack, but this semantic security can be trivially defeated under a chosen-ciphertext attack. Early versions of RSA padding used in the SSL protocol were vulnerable to a sophisticated adaptive chosen-ciphertext attack which revealed SSL session keys. Chosen-ciphertext attacks have implications for some self-synchronizing stream ciphers as well. Designers of tamper-resistant cryptographic smart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Public-key Cryptosystem
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security. There are many kinds of public-key cryptosystems, with different security goals, including digital signature, Diffie–Hellman key exchange, public-key key encapsulation, and public-key encryption. Public key algorithms are fundamental security primitives in modern cryptosystems, including applications and protocols that offer assurance of the confidentiality and authenticity of electronic communications and data storage. They underpin numerous Internet standards, such as Transport Layer Security (TLS), SSH, S/MIME, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
