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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 .
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.
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under ...
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. [1]
The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...
The GM cryptosystem is semantically secure based on the assumed intractability of the quadratic residuosity problem modulo a composite N = pq where p, q are large primes.This assumption states that given (x, N) it is difficult to determine whether x is a quadratic residue modulo N (i.e., x = y 2 mod N for some y), when the Jacobi symbol for x is +1.
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.
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.
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related to: congruence and matrices in cryptography theory 6th ed