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t 3 = 6 is the modular multiplicative inverse of 5 × 7 (mod 11). Thus, X = 3 × (7 × 11) × 4 + 6 × (5 × 11) × 4 + 6 × (5 × 7) × 6 = 3504. and in its unique reduced form X ≡ 3504 ≡ 39 (mod 385) since 385 is the LCM of 5,7 and 11. Also, the modular multiplicative inverse figures prominently in the definition of the Kloosterman sum.
When R is a power of a small positive integer b, N′ can be computed by Hensel's lemma: The inverse of N modulo b is computed by a naïve algorithm (for instance, if b = 2 then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of b, stopping when the inverse modulo R is known; N′ is ...
Let R be a ring. R is a free module of rank one over itself (either as a left or right module); any unit element is a basis. More generally, If R is commutative, a nonzero ideal I of R is free if and only if it is a principal ideal generated by a nonzerodivisor, with a generator being a basis. [3]
The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...
Inversive congruential generators are a type of nonlinear congruential pseudorandom number generator, which use the modular multiplicative inverse (if it exists) to generate the next number in a sequence. The standard formula for an inversive congruential generator, modulo some prime q is:
) To prove that the backward direction + + is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix Y {\displaystyle Y} (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix X {\displaystyle X} (in this case A + u v T {\displaystyle A+uv^{\textsf {T}}} ) if ...
In the construction of the tensor product over a commutative ring R, the R-module structure can be built in from the start by forming the quotient of a free R-module by the submodule generated by the elements given above for the general construction, augmented by the elements r ⋅ (m ∗ n) − m ∗ (r ⋅ n).
r ← (p m − 1)/(p − 1) compute A r−1 in GF(p m) compute A r = A r−1 · A; compute (A r) −1 in GF(p) compute A −1 = (A r) −1 · A r−1; return A −1; This algorithm is fast because steps 3 and 5 both involve operations in the subfield GF(p). Similarly, if a small value of p is used, a lookup table can be used for inversion in ...