Essays/NAF

The Non-adjacent_form (NAF) of a number is a unique signed-digit representation. It is useful when building scalar multipliers of Elliptic Curve points, for which the cost of doubling the point is significantly less, compared to the cost of point addition and subtraction. There can be other applications as well, e.g. for efficiently implementing an algebra with similar relative costs of operations.

In J the NAF can be built with the verb

   NAF =: (}:"1@:-/@([: #: (_1 _3)&(*/))) :. #.

It works as

   NAF 7
1 0 0 _1
   NAF i.8
0 0  0  0
0 0  0  1
0 0  1  0
0 1  0 _1
0 1  0  0
0 1  0  1
1 0 _1  0
1 0  0 _1
   |&.NAF i.8
0 1 2 5 4 5 10 9

This implementation supports array arguments and has obverse defined. Thanks to Oleg Kobchenko for array support and to Raul Miller for mentioning that #. is inverse of NAF.

Comments

You can use @SIG@ to sign your comments. This section is possibly temporary(?).

• If NAF is a replacement for #:, it should support any shapes, but

     NAF i.3
  |length error: NAF
  |       NAF i.3
     #:i.3
  0 0
  0 1
  1 0

  One solution is

     NAF2=: 13 : '}:"1-/#:_1 _3*/ y.'
     NAF2
  [: }:"1 [: -/ [: #: _1 _3"_ */ ]

  Example

     (,: #.&.>) (NAF2 ; #:)i.3 4
  +------------+---------+
  |0 0  0  0  0|0 0 0 0  |
  |0 0  0  0  1|0 0 0 1  |
  |0 0  0  1  0|0 0 1 0  |
  |0 0  1  0 _1|0 0 1 1  |
  |            |         |
  |0 0  1  0  0|0 1 0 0  |
  |0 0  1  0  1|0 1 0 1  |
  |0 1  0 _1  0|0 1 1 0  |
  |0 1  0  0 _1|0 1 1 1  |
  |            |         |
  |0 1  0  0  0|1 0 0 0  |
  |0 1  0  0  1|1 0 0 1  |
  |0 1  0  1  0|1 0 1 0  |
  |1 0 _1  0 _1|1 0 1 1  |
  +------------+---------+
  |0 1  2  3   |0 1  2  3|
  |4 5  6  7   |4 5  6  7|
  |8 9 10 11   |8 9 10 11|
  +------------+---------+

  -- OlegKobchenko 2005-11-02 21:17:42

• The inverse of NAF2 (and NAF) is #. -- Raul Miller 2005-11-03 00:52:28

Contributed by Konstantin Metlov.