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The RSA problem is defined as the task of taking e th roots modulo a composite n: recovering a value m such that c ≡ m e (mod n), where (n, e) is an RSA public key, and c is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus n .
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
More specifically, the RSA problem is to efficiently compute P given an RSA public key (N, e) and a ciphertext C ≡ P e (mod N). The structure of the RSA public key requires that N be a large semiprime (i.e., a product of two large prime numbers), that 2 < e < N, that e be coprime to φ(N), and that 0 ≤ C < N.
At every step multiplying the result from the previous iteration, c, by b and performing a modulo operation on the resulting product, thereby keeping the resulting c a small integer. The example b = 4, e = 13, and m = 497 is presented again. The algorithm performs the iteration thirteen times: (e′ = 1) c = (4 ⋅ 1) mod 497 = 4 mod 497 = 4
RSA Laboratories stated: "Now that the industry has a considerably more advanced understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active." [6] When the challenge ended in 2007, only RSA-576 and RSA-640 had been factored from the 2001 challenge numbers. [7]
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem, one of the oldest widely used for secure data transmission. The initialism "RSA" comes from the surnames of Ron Rivest , Adi Shamir and Leonard Adleman , who publicly described the algorithm in 1977.
To implement the algorithm that is described above, one should first remark that only the two last values of the indexed variables are needed at each step. Thus, for saving memory, each indexed variable must be replaced by just two variables. For simplicity, the following algorithm (and the other algorithms in this article) uses parallel ...
Example of a Key Derivation Function chain as used in the Signal Protocol.The output of one KDF function is the input to the next KDF function in the chain. In cryptography, a key derivation function (KDF) is a cryptographic algorithm that derives one or more secret keys from a secret value such as a master key, a password, or a passphrase using a pseudorandom function (which typically uses a ...