Difficult RNS Operations

1
Difficult RNS Operations

• Sign Test
• Magnitude Comparison
• Overflow Detection
• Generalized Division

Suffices to discuss first three in context of
being able todo magnitude comparison since they
are essentially same if M is such that MNP1
where the values representedare in interval
-N,P.
2
Difficult RNS Operations

• Sign Test same as Comparison with P
• Overflow Detection accomplished using Signs of
  Operands and Results
• Focus On
• Magnitude Comparison
• Generalized Division

3
Magnitude Comparison

• Could Convert to Weighted Representation Using
  CRT
• Too Complicated too much Overhead
• Use Approximate CRT Instead
• Divide CRT Equality by M

by Definition
4
Approximate CRT

• Addition of Terms is Modulo-1
• All mi-1lt?iyigtmi Are in 0,1)
• Whole Part of Result Discarded and Fractional
  Part Kept
• Much Easier than CRT Modulo-M Addition
• mi-1lt?iyigtmi Can be Precomputed for all y and i
• Use Table Lookup Circuit and Fractional
  Adder (ignore carry-outs)

5
Approximate CRT LUT
6
Magnitude Comparison Example
Use approximate CRT decoding to determine the
larger of the two numbers.
Reading the Values from the Tables
Thus, we conclude that
7
Approximate CRT Error
If Maximum Error in Approximate CRT Table is ?,
then Approximate CRT Decoding Yields Scaled Value
of RNS Number with Error No Greater than k ?
Previous Example Table Entries Rounded to 4 Digits
Maximum Error in Each Entry is ?0.00005
k4 Digits Error is 4?0.0002
0.0571-0.05360.0035 gt 4?0.0002, so XgtY is Safe
8
Redundant RNS Representations

• Do Not Have Restrict Digits in RNS to Set 0, mi
  -1
• If 0, ?i Where ?i ? mi Then RNS is Redundant
• Redundant RNS Simplifies Modular Reduction Step
  for Each Arithmetic Operation

9
Redundant RNS Example

• Consider mod-13 with 0,15
• Redundant since

• Addition Using Pseudo-redundancies Can be Done
  with Two 4-bit Adders

X
Y
Cout
00
Ignore
SUM