Methods and apparatus for modular reduction circuits
Techniques are provided for performing modular arithmetic on a key composed of many bits. One circuit implementation includes a distributor, one or more lookup tables and a plurality of adders. The distributor segments the key into a plurality of partitions. Each partition is based on a polynomial expression corresponding to a fixed size key. Each of the bits contained within the partitions are routed on a partition basis to one or more lookup tables, the routed bits acting as indices into the one or more tables. The lookup tables store precomputed values based upon the polynomial expression. The outputted precomputed values from one or more lookup tables are outputted to the plurality of adders. The plurality of adders add the bits from a portion of the routed partitions and the outputted precomputed values from the one or more lookup tables to form the binary residue.
1. A modular reduction circuit for reducing a binary key having a plurality of bits into a binary residue, the circuit comprising:
a distributor for receiving the binary key, the distributor partitions the plurality of bits into a plurality of partitions having first and second portions, the distributor partitions the plurality of bits based on a polynomial expression;
one or more lookup tables receiving the first portion of partitions, the one or more lookup tables containing precomputed values; and
a plurality of adders adding the bits of the second portion of partitions and outputted precomputed values to form the binary residue.
2. The circuit of claim 1 wherein the polynomial expression expresses the key reduction as (s mod m+qt mod m+r2 14 mod m+p2 22 mod m)(mod m).
3. The circuit of claim 1 wherein the precomputed values are based on the combinatorial values of the first portion of partitions multiplied by term constants found in the polynomial expression.
4. The circuit of claim 1 wherein the plurality of adders further comprise a plurality of 4:2 compressors and a final adder, the plurality of 4:2 compressors performing bit by bit addition on four inputs to produce an intermediate sum and a carry, the final adder summing the intermediate sum and the carry to form the binary residue.
5. The circuit of claim 1 further comprising a second modular reduction circuit to further reduce the residue.
6. The circuit of claim 1 wherein the key size is at least 32 bits and the residue has a size of at least 15 bits.
7. A method for reducing a binary key having a plurality of bits into a binary residue, the method comprising:
segmenting the plurality of bits into a plurality of partitions having first and second portions;
routing the plurality of partitions based on a polynomial expression;
retrieving precomputed values stored in a table indexed by the first portion of partitions; and
summing the second portion of partitions and retrieved precomputed values to sum terms of the polynomial expression to form the binary residue.
8. The method of claim 7 wherein the polynomial expression expresses the key reduction in mathematical form.
9. The method of claim 7 wherein the precomputed values are based on the combinatorial values of the first portion of partitions multiplied by term constants in the polynomial expression.
10. The method of claim 7 wherein the summation step further comprises:
summing the second portion of partitions; and
summing the retrieved precomputed values by a plurality of 4:2 compressors.
11. The method of claim 7 further comprising:
reducing the residue further to a final residue by outputting the residue to a modular reduction circuit.
12. The method of claim 11 wherein the key size is 32 bits and the residue has a size of at least 15 bits.
13. A modular reduction circuit for reducing a binary key having a plurality of bits into a binary residue, the circuit comprising:
means for segmenting the plurality of bits into a plurality of partitions having first and second portions based on a polynomial expression;
means for retrieving precomputed values stared in a table indexed by the first portion of partitions; and
means for summing the bits of the second portion of partitions and retrieved precomputed values to form the binary residue.
14. The circuit of claim 13 wherein the polynomial expression expresses the key reduction in mathematical form.
15. The circuit of claim 13 wherein the precomputed values are based on the combinatorial values of the first portion of partitions multiplied by term constants in the polynomial expression.
16. The circuit of claim 13 wherein the means for summing further comprises:
means for summing the second portion of partitions; and
means for summing the retrieved precomputed values.
17. The circuit of claim 13 further comprising:
means for reducing the residue further to a final residue by outputting the residue to a modular reduction circuit.
18. The circuit of claim 17 wherein the key size is at least 32 bits and the residue has a size of at least 15 bits.
19. A high differential modular reduction circuit to reduce a key having a plurality of bits to a residue, the circuit comprising:
a distributor for receiving the plurality of bits, the distributor partitions the plurality of bits into a plurality of segments;
one or more modular reduction circuits, each of the one or more modular reduction circuits receives one of the plurality of segments, respectively, each of the one or more modular reduction circuits reducing the plurality of segments, respectively, each of the one or more modular reduction circuits outputting a plurality of intermediate residues, the plurality of intermediate residues having a first and second portion;
one or more lookup tables receiving the first portion, the one or more lookup tables containing precomputed values, one or more lookup tables outputting the precomputed values to a plurality of bit streams based on a polynomial expression;
a plurality of compressors for compressing the plurality of bit streams by adding bits in the same position across the plurality of bit streams to result in at least two bits for each bit position found in the plurality of bit streams; each of the resulting at least two bits for each bit position are outputted to at least two compressor output streams; and
a final adder for summing the at least two compressor output streams.
20. The circuit of claim 19
wherein each segment in the plurality of segments have an equal number of bits,
wherein one or more modular reduction circuits have the same intermediate differential between dividend and residue.
21. The circuit of claim 19 wherein the polynomial expression expresses the key reduction in mathematical form.
22. The circuit of claim 19 wherein the precomputed values are based on the combinatorial values of the first portion of segments multiplied by term constants in the polynomial expression.
23. The circuit of claim 19 further comprising a second modular reduction circuit to further reduce the residue.
24. The circuit of claim 19 wherein the key size is at least 128 bits and the residue has a size at least 15 bits.
25. The circuit of claim 24 wherein the one or more modular reduction circuits process inputs of at least 32 bits.