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This method is an efficient variant of the 2 k-ary method. For example, to calculate the exponent 398, which has binary expansion (110 001 110) 2, we take a window of length 3 using the 2 k-ary method algorithm and calculate 1, x 3, x 6, x 12, x 24, x 48, x 49, x 98, x 99, x 198, x 199, x 398.
The operation (v & -v) zeros all bits except the least-significant bit set, resulting in a new value which is a power of 2. This power of 2 is multiplied (arithmetic modulo 2 32 ) by the de Bruijn sequence, thus producing a 32-bit product in which the bit sequence of the 5 MSBs is unique for each power of 2.
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.
is constant-recursive because it satisfies the linear recurrence = +: each number in the sequence is the sum of the previous two. [2] Other examples include the power of two sequence ,,,,, …, where each number is the sum of twice the previous number, and the square number sequence ,,,,, ….
In binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2 −1) is a right arithmetic shift, a (0) results in no operation (since 2 0 = 1 is the multiplicative identity element), and a (2 1) results in a left arithmetic shift ...
Two to the power of n, written as 2 n, is the number of values in which the bits in a binary word of length n can be set, where each bit is either of two values. A word, interpreted as representing an integer in a range starting at zero, referred to as an "unsigned integer", can represent values from 0 (000...000 2) to 2 n − 1 (111...111 2) inclusively.
The basic principle of Karatsuba's algorithm is divide-and-conquer, using a formula that allows one to compute the product of two large numbers and using three multiplications of smaller numbers, each with about half as many digits as or , plus some additions and digit shifts.
The problem here is that the low-order bits of a linear congruential PRNG with modulo 2 e are less random than the high-order ones: [6] the low n bits of the generator themselves have a period of at most 2 n. When the divisor is a power of two, taking the remainder essentially means throwing away the high-order bits, such that one ends up with ...