@MISC{RADIX_1.MIZ,
  AUTHOR = {Fujisawa, Yoshinori and Fuwa, Yasushi},
  TITLE = {Definitions of Radix-$2^k$ Signed-Digit Number and its Adder Algorithm},
  SECTION1 = {Some Useful Theorems},
  SECTION2 = {Definition for Radix-$2^k$, $k$-SD},
  SECTION3 = {Functions for Generating Radix-$2^k$ SD Numbers from Natural Numbers 
and Natural Numbers from Radix-$2^k$ SD Numbers},
  SECTION4 = {Definition for Carry and Data Components of Addition},
  SECTION5 = {Definition for Checking whether or not a Natural Number can be Expressed 
as n Digits Radix-$2^k$ SD Number},
  SECTION6 = {Definition for Addition Operation for a High-Speed Adder Algorithm 
on Radix-$2^k$ SD Number},
  DAY = {07},
  MONTH = {September},
  YEAR = {1999},
  ADDRESS1 = {Shinshu University\\Nagano},
  ADDRESS2 = {Shinshu University\\Nagano},
  SUMMARY = {In this article, a radix-$2^k$ signed-digit number (Radix-$2^k$ SD number)
  is defined and based on it a high-speed adder algorithm is discussed. \par
  The
  processes of coding and encoding for public-key cryptograms require a
  great deal of addition operations of natural number of many figures.
  This results in a~long time for the encoding and decoding processes. It
  is possible to reduce the processing time using the high-speed adder
  algorithm.\par
  In the first section of this article, we prepared some useful theorems 
  for natural
  numbers and integers. In the second section, we defined the concept
  of radix-$2^k$, a set named $k$-SD and proved some properties about
  them. 
  In the third section, we provide some important functions for
  generating Radix-$2^k$ SD numbers from natural numbers and natural numbers
  from Radix-$2^k$ SD numbers. In the fourth section, we defined the carry
  and data components of addition with Radix-$2^k$ SD numbers and some
  properties about them. In the fifth section, we defined a theorem for
  checking whether or not a natural number can be expressed as $n$ digits
  Radix-$2^k$ SD number. \par
  In the last section, a high-speed adder algorithm
  on Radix-$2^k$ SD numbers is proposed and we provided some properties.
  In this algorithm, the carry of each digit has an effect on only the
  next digit. Properties of the relationships of the results of this
  algorithm to the operations of natural numbers are also given.}}

