Optimal Signed Binary Recoding For Integers

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Optimal Signed Binary Recoding For Integers
Xiaoyu Ruan, Rajendra Katti Department of
Electrical and Computer Engineering North Dakota
State University
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Signed Binary Recoding Overview

• Signed Binary Number System uses 0,1,1 to
  recode integers.
• SB recoding uses at least one more bit than
  binary expansion.
• An integer has many different forms of SB
  representation
• Weight the number of non-zeros in a
  representation.
• SB expansion could have lower weight than
  binary.
• Joint Weight number of non-zero columns in a SB
  table.

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Elliptic Curve Cryptosystem (ECC)

• The main operation of ECC is to compute
• where ks are keys and Ps points on elliptic
  curve.
• Using the signed binary system to recode ks
  reduces the number of non-zero columns (joint
  weight), thus speeds up the operation.
• In an ECC, the ks are received from left to
  right (MSB to LSB).
• is also computed from left to
  right using Shamirs method.
• -P is obtained for free from P. Hence SB
  expansions for ks are introduced.

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Shamirs Method in ECC

• To compute 6P15P2.
• Shamirs Method Operates from left to right (MSB
  to LSB).
• Number of operations depends on joint weight of
  kis.

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Optimal Signed Binary Recoding

• To speed up ECC, optimal SB recoding should
• (1) Result in minimum possible joint weight.
• (2) Operate from left to right.
• For a single integer, Joye and Yen introduced a
  method that satisfies both (1) and (2).
• For any number of integers, Solinas and Proos at
  Center for Cryptography Research at University of
  Waterloo introduced a method, Joint Sparse Form,
  that satisfies (1) but not (2). As a result, a
  large amount of memory is required for
  computation.
• We propose a new method, that satisfies both (1)
  and (2).

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Basic Replacement Principle
Examples
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Recoding Algorithm

• INPUT N binary integers ki. Maximum length is L
  bits.
• OUTPUT (L1)-bit SB recoding table of ki, with
  the minimum possible joint weight.
• STEP 1 (Preliminary Replacement) Left-to-right
  scan ks. Convert binary expansions to the
  alternating greedy expansion.
• For an alternating greedy expansion
• xs neighboring non-zero bit(s) must be x.
• Leftmost non-zero bit of each row is 1.
• Rightmost non-zero bit of each row is 1.
• Unique for a given integer.

k ((kL1 0), (kL2 kL1),, (k0 k1), (0
k0))
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Recoding Algorithm, Cont.

• INPUT N binary integers ki. Maximum length is L
  bits.
• OUTPUT (L1)-bit SB expansion table of ki, with
  the minimum possible joint weight.
• STEP 2 (Crucial Replacement) Left-to-right scan
  alternating greedy expansion table of ks column
  by column. Look at atmost (N1) columns at a
  time. Make appropriate replacements on the
  N(N1) table which is being examined.
• The two steps can be combined. Thus the
  intermediate SB expansion does not need to be
  stored in memory.

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Example

• Suppose our ECC receives 4 binary integers
  (keys), maximum length 11 bits. (N 4, L 11)
• STEP 1 Convert to alternating greedy
  expansions.

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Example, Cont.

• STEP 2 Make replacements on alternating greedy
  expansion using the rule
• The sequences in the same color indicate a
  replacement.
• The joint weight is 8 for binary expansion and 7
  for the final optimal output, reduced by 1.

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Properties

• Theorem 1. (Uniqueness)
• Given N non-negative integers, the SB
  expansions output by the our method is unique.
• Theorem 2. (Optimality)
• The SB expansion output by the our method has
  the minimum joint weight among any SB expansions
  of the given N integers.
• Theorem 3. (Density of Zero Columns)
• Amongst every 2N1 consecutive columns of the
  output of our method, there exists at least one
  zero column.

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Proof of Optimality

• Step 1 Take any SB expansion table of the N
  given integers with joint weight w.
• Step 2 Convert this representation to the unique
  alternating greedy expansion table with joint
  weight w1.
• Step 3 Input the unique alternating greedy
  expansion table to Step 2 of the algorithm.
  Output is with joint weight w2.
• Step 4 Show that w2 w in any case.

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New Method vs. Joint Sparse Form

• Joint Weight
• Both methods result in minimum joint weight.
• Software Implementation
• Our Method uses less CPU time than JSF Method
  does. (approximately 85)
• Hardware Implementation
• Our Method left-to-right, intermediate results
  not stored.
• Needs Memory N(N1) SB bits.
• JSF Method right-to-left, intermediate results
  must be stored.
• Needs Memory N(L1) SB bits.

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Thank you

• Questions and Comments Please