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In the asymptotic setting, a family of deterministic polynomial time computable functions : {,} {,} for some polynomial p, is a pseudorandom number generator (PRNG, or PRG in some references), if it stretches the length of its input (() > for any k), and if its output is computationally indistinguishable from true randomness, i.e. for any probabilistic polynomial time algorithm A, which ...
special designs based on mathematical hardness assumptions: examples include the Micali–Schnorr generator, [18] Naor-Reingold pseudorandom function and the Blum Blum Shub algorithm, which provide a strong security proof (such algorithms are rather slow compared to traditional constructions, and impractical for many applications)
A structure similar to LCGs, but not equivalent, is the multiple-recursive generator: X n = (a 1 X n−1 + a 2 X n−2 + ··· + a k X n−k) mod m for k ≥ 2. With a prime modulus, this can generate periods up to m k −1, so is a useful extension of the LCG structure to larger periods.
A C version [a] of three xorshift algorithms [1]: 4,5 is given here. The first has one 32-bit word of state, and period 2 32 −1. The second has one 64-bit word of state and period 2 64 −1. The last one has four 32-bit words of state, and period 2 128 −1. The 128-bit algorithm passes the diehard tests.
Blum Blum Shub takes the form + =, where M = pq is the product of two large primes p and q.At each step of the algorithm, some output is derived from x n+1; the output is commonly either the bit parity of x n+1 or one or more of the least significant bits of x n+1.
The Python code below assumes the input graph is given as a -adjacency matrix with zeros on the diagonal. It defines the function APD which returns a matrix with entries D i , j {\displaystyle D_{i,j}} such that D i , j {\displaystyle D_{i,j}} is the length of the shortest path between the vertices i {\displaystyle i} and j {\displaystyle j} .
The simple Sethi–Ullman algorithm works as follows (for a load/store architecture): . Traverse the abstract syntax tree in pre- or postorder . For every leaf node, if it is a non-constant left-child, assign a 1 (i.e. 1 register is needed to hold the variable/field/etc.), otherwise assign a 0 (it is a non-constant right child or constant leaf node (RHS of an operation – literals, values)).
The algorithm considers two problems: the master problem and the subproblem. The master problem is the original problem with only a subset of variables being considered. The subproblem is a new problem created to identify an improving variable (i.e. which can improve the objective function of the master problem). The algorithm then proceeds as ...