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Advanced Micro Devices, Inc. ( AMD) is an American multinational corporation and semiconductor company based in Santa Clara, California, that develops computer processors and related technologies for business and consumer markets. AMD's main products include microprocessors, motherboard chipsets, embedded processors and graphics processors for ...
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- History
- Patent
- Operation
- Proofs of Correctness
- Padding
- Security and Practical Considerations
- Implementations
- See Also
- Further Reading
- External Links
The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they ...
A patent describing the RSA algorithm was granted to MIT on 20 September 1983: U.S. patent 4,405,829 "Cryptographic communications system and method". From DWPI's abstract of the patent: A detailed description of the algorithm was published in August 1977, in Scientific American's Mathematical Games column. This preceded the patent's filing date of...
The RSA algorithm involves four steps: keygeneration, key distribution, encryption, and decryption. A basic principle behind RSA is the observation that it is practical to find three very large positive integers e, d, and n, such that for all integers m (0 ≤ m < n), both ( m e ) d {\displaystyle (m^{e})^{d}} and m {\displaystyle m} have the same re...
Proof using Fermat's little theorem
The proof of the correctness of RSA is based on Fermat's little theorem, stating that ap − 1 ≡ 1 (mod p) for any integer a and prime p, not dividing a.[note 1] We want to show that Since λ(pq) = lcm(p − 1, q − 1) is, by construction, divisible by both p − 1 and q − 1, we can write To check whether two numbers, such as med and m, are congruent mod pq, it suffices (and in fact is equivalent) to check that they are congruent mod p and mod q separately.[note 3] To show med ≡ m (mod p), we conside...
Proof using Euler's theorem
Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on Euler's theorem. We want to show that med ≡ m (mod n), where n = pq is a product of two different prime numbers, and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)). Since e and d are positive, we can write ed = 1 + hφ(n) for some non-negative integer h. Assuming that m is relatively prime to n, we have where the second-l...
Attacks against plain RSA
There are a number of attacks against plain RSA as described below. 1. When encrypting with low encryption exponents (e.g., e = 3) and small values of the m (i.e., m < n1/e), the result of me is strictly less than the modulus n. In this case, ciphertexts can be decrypted easily by taking the eth root of the ciphertext over the integers. 2. If the same clear-text message is sent to e or more recipients in an encrypted way, and the receivers share the same exponent e, but different p, q, and th...
Padding schemes
To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. This padding ensures that mdoes not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts. Standards such as PKCS#1 have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext m with...
Using the Chinese remainder algorithm
For efficiency, many popular crypto libraries (such as OpenSSL, Java and .NET) use for decryption and signing the following optimization based on the Chinese remainder theorem.[citation needed]The following values are precomputed and stored as part of the private key: 1. p {\displaystyle p} and q {\displaystyle q} – the primes from the key generation, 2. d P = d ( mod p − 1 ) , {\displaystyle d_{P}=d{\pmod {p-1}},} 3. d Q = d ( mod q − 1 ) , {\displaystyle d_{Q}=d{\pmod {q-1}},} 4. q inv = q...
Integer factorization and the RSA problem
The security of the RSA cryptosystem is based on two mathematical problems: the problem of factoring large numbers and the RSA problem. Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are hard, i.e., no efficient algorithm exists for solving them. Providing security against partial decryption may require the addition of a secure padding scheme. The RSA problem is defined as the task of taking eth roots modulo a composite n: recove...
Faulty key generation
Finding the large primes p and q is usually done by testing random numbers of the correct size with probabilistic primality teststhat quickly eliminate virtually all of the nonprimes. The numbers p and q should not be "too close", lest the Fermat factorization for n be successful. If p − q is less than 2n1/4 (n = p⋅q, which even for "small" 1024-bit values of n is 3×1077), solving for p and q is trivial. Furthermore, if either p − 1 or q − 1 has only small prime factors, n can be factored qui...
Some cryptography libraries that provide support for RSA include: 1. Botan 2. Bouncy Castle 3. cryptlib 4. Crypto++ 5. Libgcrypt 6. Nettle 7. OpenSSL 8. wolfCrypt 9. GnuTLS 10. mbed TLS 11. LibreSSL
Menezes, Alfred; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). Handbook of Applied Cryptography. CRC Press. ISBN 978-0-8493-8523-0.Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 881–887. ISBN 978-0-262-03293-3.The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977, U.S. patent 4,4...PKCS #1: RSA Cryptography Standard (RSA Laboratories website)Explanation of RSA using colored lamps on YouTubeThe Federal Reserve System (often shortened to the Federal Reserve, or simply the Fed) is the central banking system of the United States.It was created on December 23, 1913, with the enactment of the Federal Reserve Act, after a series of financial panics (particularly the panic of 1907) led to the desire for central control of the monetary system in order to alleviate financial crises.
Yasuke ( Japanese: 弥助 / 弥介, Japanese pronunciation: [jasɯ̥ke]) was a man of African origin [2] [3] who served as a retainer to the Japanese daimyō Oda Nobunaga for a period of 15 months between 1581 and 1582, during the Sengoku period, until Nobunaga's death in the Honnō-ji Incident. [4] [5] There are few historical documents on Yasuke.
Definition. The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling (see preceding image) The Fibonacci numbers may be defined by the recurrence relation [6] and for n > 1 .
Ludwig van Beethoven [n 1] (baptised 17 December 1770 – 26 March 1827) was a German composer and pianist. He is one of the most revered figures in the history of Western music; his works rank among the most performed of the classical music repertoire and span the transition from the Classical period to the Romantic era in classical music.
