Ads
related to: integer relation algorithm worksheet practice sheet key answers
Search results
Results from the WOW.Com Content Network
An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain upper bound .
LLL is widely used in the cryptanalysis of public key cryptosystems. When used to find integer relations, a typical input to the algorithm consists of an augmented identity matrix with the entries in the last column consisting of the elements (multiplied by a large positive constant to penalize vectors that do not sum to zero) between which the ...
An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture. [5]The LLL algorithm has found numerous other applications in MIMO detection algorithms [6] and cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, NTRUEncrypt, and so forth.
This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve. The size of the input to the algorithm is log 2 n or the number of bits in the binary representation of n. Any element of the order n c for a constant c is exponential in log n.
[3] [6] [7] I.e., if there exists an algorithm that can efficiently break the cryptographic scheme with non-negligible probability, then there exists an efficient algorithm that solves a certain lattice problem on any input. However, for the practical lattice-based constructions (such as schemes based on NTRU and even schemes based on LWE with ...
The property is the following. For a given odd integer n > 2, let’s write n − 1 as 2 s d where s is a positive integer and d is an odd positive integer. Let’s consider an integer a, called a base, which is coprime to n. Then, n is said to be a strong probable prime to base a if one of these congruence relations holds:
ECM is at its core an improvement of the older p − 1 algorithm. The p − 1 algorithm finds prime factors p such that p − 1 is b-powersmooth for small values of b. For any e, a multiple of p − 1, and any a relatively prime to p, by Fermat's little theorem we have a e ≡ 1 (mod p). Then gcd(a e − 1, n) is likely to produce a factor of n.
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 Ajtai [ 1 ] who presented a family of one-way functions based on SIS problem.
Ads
related to: integer relation algorithm worksheet practice sheet key answers