Топ-100 ★ Non-adjacent form - non-standard positional numeral syste

★ Non-adjacent form - non-standard positional numeral systems ..

Non-adjacent form

★ Non-adjacent form

The non-adjacent form of a number is a unique signed-digit representation. Like the name suggests, non-zero values cannot be adjacent. For example:

0 1 2 = 4 + 2 + 1 = 7 1 0 −1 2 = 8 − 2 + 1 = 7 −1 2 = 8 − 4 + 2 + 1 = 7 1 0 −1 2 = 8 − 1 = 7

All right signed-digit representations of 7, but only in the final representation, 1 0 -1 2, NAF.


1. Properties. (Свойства)

NAF provides a unique view of the whole, but the main advantage of it is that Hamming weight of value will be minimal. For regular binary representations of values, half of all bits will not be zero, on average, but with NAF this drops to only a third of all numbers.

It is obvious that more than half of the numbers, different from zero, which was the reason it was introduced by V. G. Reitweisner to accelerate algorithms in the beginning of the multiplication, as well as coding Booth.

Because each non-zero digit must be adjacent to two 0 C, NAF representation can be implemented so that it only accepts a maximum of M 1 bits per value, which must be represented in binary with m bits.

NAF properties make it useful in various algorithms, especially in cryptography, for example, to reduce the number of multiplications required to perform exponentiation. In the algorithm of fast exponentiation, the number of multiplication depends on the number of nonzero bits. If the exponent here is given in NAF form, a digit value of 1 means multiplying by the base, and a digital value of -1 for its return.

Other ways of coding integers that are not consecutive 1S includes coding the stand and encode Fibonacci.


2. Converting to NAF. (Преобразование в НВС)

There are several algorithms for obtaining the NAF representation of the value of this in binary. One such is the following method using repeated division, it works by selecting non-zero coefficients so that the resulting quotient is divisible by 2 and therefore, the next coefficient is zero.

Input E = e m − 1 e m − 2 e 1 e 0 2 Output Z = z m z m − 1 z 1 z 0 NAF i ← 0 while E >, 0 do if E is odd then z i ← 2 − E mod 4 E ← E − z i else z i ← 0 E ← E /2 i ← i + 1 return z
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