A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two
I present a new algorithm for computing binomial coefficients modulo . The proposed method has an preprocessing time, after which a binomial coefficient with can be computed modulo in time. denotes the time complexity of multiplying two -bit numbers, which can range from to or better. Thus,...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2013-01-01
|
| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2013/751358 |
| Summary: | I present a new algorithm for computing binomial coefficients modulo . The proposed method has an preprocessing time, after which a binomial coefficient with can be computed modulo in time. denotes the time complexity of multiplying two -bit numbers, which can range from to or better. Thus, the
overall time complexity for evaluating binomial coefficients modulo with is . After preprocessing, we can actually compute binomial coefficients modulo any with . For larger values of and , variations
of Lucas’ theorem must be used first in order to reduce the computation to
the evaluation of multiple binomial coefficients (or restricted types of factorials ) modulo with . |
|---|---|
| ISSN: | 1537-744X |