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An MDS matrix (maximum distance separable) is a matrix representing a function with certain diffusion properties that have useful applications in cryptography.Technically, an matrix over a finite field is an MDS matrix if it is the transformation matrix of a linear transformation = from to such that no two different (+)-tuples of the form (, ()) coincide in or more components.
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 .
Secret sharing consists of recovering a secret S from a set of shares, each containing partial information about the secret. The Chinese remainder theorem (CRT) states that for a given system of simultaneous congruence equations, the solution is unique in some Z/nZ, with n > 0 under some appropriate conditions on the congruences.
In cryptography, confusion and diffusion are two properties of a secure cipher identified by Claude Shannon in his 1945 classified report A Mathematical Theory of Cryptography. [1] These properties, when present, work together to thwart the application of statistics , and other methods of cryptanalysis .
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even .
It is easy to show that the trace of a matrix representing an element of Γ(N) cannot be −1, 0, or 1, so these subgroups are torsion-free groups. (There are other torsion-free subgroups.) The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2, Z/2Z) is isomorphic to S 3, Λ is a subgroup of index 6.
In strong cryptography, b is often at least 1024 bits. [1] Consider b = 5 × 10 76 and e = 17, both of which are perfectly reasonable values. In this example, b is 77 digits in length and e is 2 digits in length, but the value b e is 1,304 decimal digits in length. Such calculations are possible on modern computers, but the sheer magnitude of ...
In contrast to non-commutative cryptography, the currently widely used public-key cryptosystems like RSA cryptosystem, Diffie–Hellman key exchange and elliptic curve cryptography are based on number theory and hence depend on commutative algebraic structures.
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