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Rabin Cryptosystem
The Rabin cryptosystem is a family of public-key encryption schemes based on a trapdoor function whose security, like that of RSA, is related to the difficulty of integer factorization. The Rabin trapdoor function has the advantage that inverting it has been mathematically proven to be as hard as factoring integers, while there is no such proof known for the RSA trapdoor function. It has the disadvantage that each output of the Rabin function can be generated by any of four possible inputs; if each output is a ciphertext, extra complexity is required on decryption to identify which of the four possible inputs was the true plaintext. Naive attempts to work around this often either enable a chosen-ciphertext attack to recover the secret key or, by encoding redundancy in the plaintext space, invalidate the proof of security relative to factoring. Public-key encryption schemes based on the Rabin trapdoor function are used mainly for examples in textbooks. In contrast, RSA is the basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Public-key Encryption
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. In a public-key encryption system, anyone with a public key can encrypt a message, yielding a ciphertext, but only those who know the corresponding private key can decrypt the ciphertext to obtain the original message. For example, a journalist can publish the public key of an encryption key pair on a web site so that sources can send secret messages to the news organization in ciphertext. Only the journalist who knows the corresponding private key can decrypt the ciphertexts to obtain the sources' messages—an eavesdropp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptographic Hash Function
A cryptographic hash function (CHF) is a hash algorithm (a map of an arbitrary binary string to a binary string with fixed size of n bits) that has special properties desirable for cryptography: * the probability of a particular n-bit output result (hash value) for a random input string ("message") is 2^ (like for any good hash), so the hash value can be used as a representative of the message; * finding an input string that matches a given hash value (a ''pre-image'') is unfeasible, unless the value is selected from a known pre-calculated dictionary (" rainbow table"). The ''resistance'' to such search is quantified as security strength, a cryptographic hash with n bits of hash value is expected to have a ''preimage resistance'' strength of n bits. A ''second preimage'' resistance strength, with the same expectations, refers to a similar problem of finding a second message that matches the given hash value when one message is already known; * finding any pair of different messa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blum–Goldwasser Cryptosystem
The Blum–Goldwasser (BG) cryptosystem is an asymmetric key encryption algorithm proposed by Manuel Blum and Shafi Goldwasser in 1984. Blum–Goldwasser is a probabilistic, semantically secure cryptosystem with a constant-size ciphertext expansion. The encryption algorithm implements an XOR-based stream cipher using the Blum-Blum-Shub (BBS) pseudo-random number generator to generate the keystream. Decryption is accomplished by manipulating the final state of the BBS generator using the private key, in order to find the initial seed and reconstruct the keystream. The BG cryptosystem is semantically secure based on the assumed intractability of integer factorization; specifically, factoring a composite value N = pq where p, q are large primes. BG has multiple advantages over earlier probabilistic encryption schemes such as the Goldwasser–Micali cryptosystem. First, its semantic security reduces solely to integer factorization, without requiring any additional assumptions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schmidt–Samoa Cryptosystem
The Schmidt-Samoa cryptosystem is an asymmetric cryptographic technique, whose security, like Rabin depends on the difficulty of integer factorization. Unlike Rabin this algorithm does not produce an ambiguity in the decryption at a cost of encryption speed. Key generation * Choose two large distinct primes ''p'' and ''q'' and compute N = p^2q * Compute d = N^ \mod \text(p-1,q-1) Now ''N'' is the public key and ''d'' is the private key. Encryption To encrypt a message ''m'' we compute the ciphertext as c = m^N\mod N. Decryption To decrypt a ciphertext ''c'' we compute the plaintext as m = c^d \mod pq, which like for Rabin and RSA can be computed with the Chinese remainder theorem. Example: * p = 7, q = 11, N = p^2q = 539, d = N^ \mod \text(p-1,q-1) = 29 * m = 32, c = m^N \mod N = 373 Now to verify: * m = c^d \mod pq = 373^ \mod pq = 373^ \mod 77 = 32 Security The algorithm, like Rabin, is based on the difficulty of factoring the modulus ''N'', which is a distinct advant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blum Blum Shub
Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub that is derived from Michael O. Rabin's one-way function. __TOC__ Blum Blum Shub takes the form :x_ = x_n^2 \bmod M, where ''M'' = ''pq'' is the product of two large primes ''p'' and ''q''. At each step of the algorithm, some output is derived from ''x''''n''+1; the output is commonly either the bit parity of ''x''''n''+1 or one or more of the least significant bits of ''x''''n''+1''. The seed ''x''0 should be an integer that is co-prime to ''M'' (i.e. ''p'' and ''q'' are not factors of ''x''0) and not 1 or 0. The two primes, ''p'' and ''q'', should both be congruent to 3 (mod 4) (this guarantees that each quadratic residue has one square root which is also a quadratic residue), and should be safe primes with a small gcd((''p-3'')''/2'', (''q-3'')''/2'') (this makes the cycle length large). An interesting characteristic of the Blum Blum Shub generator is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topics In Cryptography
The following outline is provided as an overview of and topical guide to cryptography: Cryptography (or cryptology) – practice and study of hiding information. Modern cryptography intersects the disciplines of mathematics, computer science, and engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce. Essence of cryptography * Cryptographer * Encryption/decryption * Cryptographic key * Cipher * Ciphertext * Plaintext * Code * Tabula recta * Alice and Bob Uses of cryptographic techniques * Commitment schemes * Secure multiparty computation * Electronic voting * Authentication * Digital signatures * Crypto systems * Dining cryptographers problem * Anonymous remailer * Pseudonymity * Onion routing * Digital currency * Secret sharing * Indistinguishability obfuscation Branches of cryptography * Multivariate cryptography * Post-quantum cryptography * Quantum cryptography * Steganography * Visual cryptography History ... [...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 hidden 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 sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chosen Plaintext
A chosen-plaintext attack (CPA) is an attack model for cryptanalysis which presumes that the attacker can obtain the ciphertexts for arbitrary plaintexts.Ross Anderson, ''Security Engineering: A Guide to Building Dependable Distributed Systems''. The first edition (2001): http://www.cl.cam.ac.uk/~rja14/book.html The goal of the attack is to gain information that reduces the security of the encryption scheme. Modern ciphers aim to provide semantic security, also known as ''ciphertext indistinguishability under chosen-plaintext attack'', and they are therefore, by design, generally immune to chosen-plaintext attacks if correctly implemented. Introduction In a chosen-plaintext attack the adversary can (possibly adaptively) ask for the ciphertexts of arbitrary plaintext messages. This is formalized by allowing the adversary to interact with an encryption oracle, viewed as a black box. The attacker’s goal is to reveal all or a part of the secret encryption key. It may seem infea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ciphertext Indistinguishability
Ciphertext indistinguishability is a property of many encryption schemes. Intuitively, if a cryptosystem possesses the property of indistinguishability, then an adversary will be unable to distinguish pairs of ciphertexts based on the message they encrypt. The property of indistinguishability under chosen plaintext attack is considered a basic requirement for most provably secure public key cryptosystems, though some schemes also provide indistinguishability under chosen ciphertext attack and adaptive chosen ciphertext attack. Indistinguishability under chosen plaintext attack is equivalent to the property of semantic security, and many cryptographic proofs use these definitions interchangeably. A cryptosystem is considered ''secure in terms of indistinguishability'' if no adversary, given an encryption of a message randomly chosen from a two-element message space determined by the adversary, can identify the message choice with probability significantly better than that of rand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer (the base) is raised to the power (the exponent), and divided by a positive integer (the modulus); that is, . From the definition of division, it follows that . For example, given , and , dividing by leaves a remainder of . Modular exponentiation can be performed with a ''negative'' exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and . Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent when given , , and – is believed to be difficult. This one-way function behavior makes modular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
