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It is a form of partitioning cryptanalysis that exploits unevenness in how the cipher operates over equivalence classes (congruence classes) modulo n. The method was first suggested in 1999 by John Kelsey , Bruce Schneier , and David Wagner and applied to RC5P (a variant of RC5 ) and M6 (a family of block ciphers used in the FireWire standard).
Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space : two matrices are congruent if and only if they represent the same bilinear form with respect to different bases .
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.
Calculating this solution by standard linear algebra algorithms then takes very little time. While matrix multiplication alone does not result in a secure cipher it is still a useful step when combined with other non-linear operations, because matrix multiplication can provide diffusion. For example, an appropriately chosen matrix can guarantee ...
Inverting the encryption function can be shown to be as hard as factoring n, meaning that if an adversary could recover the entire message from the encryption of the message they would be able to factor n.
The HFE family of cryptosystems is based on the hardness of the problem of finding solutions to a system of multivariate quadratic equations (the so-called MQ problem) since it uses private affine transformations to hide the extension field and the private polynomials. Hidden Field Equations also have been used to construct digital signature ...
Coppersmith’s algorithm can be used to find this integer solution . Finding roots over Q is easy using, e.g., Newton's method , but such an algorithm does not work modulo a composite number M . The idea behind Coppersmith’s method is to find a different polynomial f related to F that has the same root x 0 {\displaystyle x_{0}} modulo M ...
The algorithm attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n.The algorithm works in two phases: the data collection phase, where it collects information that may lead to a congruence of squares; and the data processing phase, where it puts all the data it has collected into a matrix and solves it to obtain a ...