System and method for reducing the computation and storage requirements for a montgomery-style reduction
View Patent ↗A system and method are described that provide an alternative way in which to produce a Montgomery reduction from below by storing a new precomputed value used to substantially replace the μ and n values used in Montgomery reduction with a single value. By modifying the Montgomery reduction mechanism in this way, the number of multiplications and registers required to effect the Montgomery reduction can be reduced. To avoid having to store both μ and n, a modified reduction value or a logical shift or signed version of such a value can be used in place of μ and n for the bulk of the low-order reduction.
1. A method for performing, on a cryptographic apparatus, a Montgomery-style reduction in a cryptographic operation, the method comprising:
obtaining an operand for the cryptographic operation;
computing a modified operand using a reduction value, instead of a modulus used in performing a standard Montgomery reduction, to perform a replacement of a least significant word of the operand, rather than perform a cancellation thereof, the reduction value being a function of the modulus; and
outputting the modified operand.
2. The method according to claim 1 wherein the reduction value is n′=2 −w mod n, or a shifted or signed version of n′, w corresponds to a word size, and n corresponds to the modulus.
3. The method according to claim 1 , wherein the computing further comprises:
successively applying the reduction value to perform a replacement of each of the second least significant word of the operand through the second most significant word of the operand; and
performing a standard Montgomery reduction on the most significant word of the operand.
4. The method according to claim 3 , wherein the performing a standard Montgomery reduction comprises storing a precomputed value μ in a register, using the value μ in computing another value m, and overwriting the register with m.
5. The method according to claim 1 , wherein the cryptographic apparatus comprises a Montgomery engine configured to perform the cryptographic operation.
6. The method according to claim 1 , wherein the reduction value is pre-computed and stored with one or more cryptographic system parameters prior to the computing.
7. The method according to claim 1 , wherein the performing comprises zeroing the least significant word of the operand, modifying one or more remaining words, and shifting one or more modified words, wherein the shifting is either logical or physical.
8. The method according to claim 7 , wherein if a carry is produced during the computing, the outputting comprises adding the carry as a most significant word in the modified operand.
9. The method according to claim 1 , wherein said cryptographic operation comprises multiplication or squaring.
10. A cryptographic apparatus comprising a processor configured to operate as a Montgomery engine, and computer executable instructions that when executed by the processor:
obtain an operand for the cryptographic operation;
compute a modified operand using a reduction value, instead of a modulus used in performing a standard Montgomery reduction, to perform a replacement of a least significant word of the operand, rather than perform a cancellation thereof, the reduction value being a function of the modulus; and
output the modified operand.
11. The apparatus according to claim 10 , wherein said reduction value is n′=2 −w mod n, or a shifted or signed version of n′, w corresponds to a word size, and n corresponds to the modulus.
12. The apparatus according to claim 10 , wherein the computing further comprises:
successively applying the reduction value to perform a replacement of each of the second least significant word of the operand through the second most significant word of the operand; and
performing a standard Montgomery reduction on the most significant word of the operand.
13. The apparatus according to claim 12 , wherein performing a standard Montgomery reduction comprises storing a precomputed value μ in a register, using the value μ in computing another value m, and overwriting the register with m.
14. The apparatus according to claim 10 , wherein the reduction value is pre-computed and stored with one or more cryptographic system parameters prior to the computing.
15. The apparatus according to claim 10 , wherein the performing comprises zeroing the least significant word of the operand, modifying one or more remaining words, and shifting one or more modified words, wherein the shifting is either logical or physical.
16. The apparatus according to claim 15 , wherein if a carry is produced during the computing, the apparatus is configured to add the carry as a most significant word in the modified operand.
17. The apparatus according to claim 10 , wherein the cryptographic operation comprises multiplication or squaring.
18. A non-transitory computer readable medium comprising computer executable instructions that when executed by a cryptographic apparatus, cause the cryptographic apparatus to:
obtain an operand for the cryptographic operation;
compute a modified operand using a reduction value, instead of a modulus used in performing a standard Montgomery reduction, to perform a replacement of a least significant word of the operand, rather than perform a cancellation thereof, the reduction value being a function of the modulus; and
output the modified operand.
19. The non-transitory computer readable medium according to claim 18 , wherein said reduction value is n′=2 −w mod n, or a shifted or signed version of n′, w corresponds to a word size and n corresponds to the modulus.
20. The non-transitory computer readable medium according to claim 18 , wherein the computing further comprises:
successively applying the reduction value to perform a replacement of each of the second least significant word of the operand through the second most significant word of the operand; and
performing a standard Montgomery reduction on the most significant word of the operand.
21. The non-transitory computer readable medium according to claim 20 , wherein performing a standard Montgomery reduction comprises storing a precomputed value μ in a register, using the value μ in computing another value m, and overwriting the register with m.
22. The non-transitory computer readable medium according to claim 18 , wherein the reduction value is pre-computed and stored with one or more cryptographic system parameters prior to the computing.
23. The non-transitory computer readable medium according to claim 18 , wherein the performing comprises zeroing the least significant word of the operand, modifying one or more remaining words, and shifting one or more modified words, wherein the shifting is either logical or physical.
24. The non-transitory computer readable medium according to claim 23 , wherein if a carry is produced during the computing, executing instructions to add the carry as a most significant word in the modified operand.
25. The non-transitory computer readable medium according to claim 18 , wherein the cryptographic operation comprises multiplication or squaring.