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A straightforward algorithm to multiply numbers in Montgomery form is therefore to multiply aR mod N, bR mod N, and R′ as integers and reduce modulo N. For example, to multiply 7 and 15 modulo 17 in Montgomery form, again with R = 100, compute the product of 3 and 4 to get 12 as above.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. That is, if A is an m × n matrix and B is an s × p matrix, then n needs to be equal to s for the matrix product AB to be defined.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
Naïve matrix multiplication requires one multiplication for each "1" of the left column. Each of the other columns (M1-M7) represents a single one of the 7 multiplications in the Strassen algorithm. The sum of the columns M1-M7 gives the same result as the full matrix multiplication on the left.
Example: 100P can be written as 2(2[P + 2(2[2(P + 2P)])]) and thus requires six point double operations and two point addition operations. 100P would be equal to f(P, 100). This algorithm requires log 2 (d) iterations of point doubling and addition to compute the full point multiplication. There are many variations of this algorithm such as ...
A Montgomery curve of equation = + +. A Montgomery curve over a field K is defined by the equation,: = + + for certain A, B ∈ K and with B(A 2 − 4) ≠ 0.. Generally this curve is considered over a finite field K (for example, over a finite field of q elements, K = F q) with characteristic different from 2 and with A ≠ ±2 and B ≠ 0, but they are also considered over the rationals with ...