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,...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2013-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2013/751358 |
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doaj-be8b75fe49a34ab4ae9522af13610e962020-11-24T21:45:14ZengHindawi LimitedThe Scientific World Journal1537-744X2013-01-01201310.1155/2013/751358751358A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of TwoMugurel Ionut Andreica0Computer Science Department, Politehnica University of Bucharest, Splaiul Independentei 313, Sector 6, 060042 Bucharest, RomaniaI 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 .http://dx.doi.org/10.1155/2013/751358 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Mugurel Ionut Andreica |
| spellingShingle |
Mugurel Ionut Andreica A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two The Scientific World Journal |
| author_facet |
Mugurel Ionut Andreica |
| author_sort |
Mugurel Ionut Andreica |
| title |
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two |
| title_short |
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two |
| title_full |
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two |
| title_fullStr |
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two |
| title_full_unstemmed |
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two |
| title_sort |
fast algorithm for computing binomial coefficients modulo powers of two |
| publisher |
Hindawi Limited |
| series |
The Scientific World Journal |
| issn |
1537-744X |
| publishDate |
2013-01-01 |
| description |
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 . |
| url |
http://dx.doi.org/10.1155/2013/751358 |
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