Sparse polynomial interpolation based on diversification
We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari (1988) for interpolating polynomials over fields with characteristic zero, we develop a new Monte Carlo algorithm over finite fi...
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| Vydané v: | Science China. Mathematics Ročník 65; číslo 6; s. 1147 - 1162 |
|---|---|
| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Beijing
Science China Press
01.06.2022
Springer Nature B.V |
| Predmet: | |
| ISSN: | 1674-7283, 1869-1862 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari (1988) for interpolating polynomials over fields with characteristic zero, we develop a new Monte Carlo algorithm over finite fields by doing additional probes. To interpolate a polynomial
f
∈
F
q
[
x
1
,
…
,
x
n
]
with a partial degree bound
D
and a term bound
T
, our new algorithm costs
O
∼
(
n
T
log
2
q
+
n
T
D
log
q
)
bit operations and uses 2(
n
+ 1)
T
probes to the black box. If
q
⩾
O
(
nT
2
D
), it has constant success rate to return the correct polynomial. Compared with the previous algorithms over general finite field, our algorithm has better complexity in the parameters
n, T
and
D
, and is the first one to achieve the complexity of fractional power about
D
, while keeping linear in
n
and
T
. A key technique is a randomization which makes all the coefficients of the unknown polynomial distinguishable, producing a diverse polynomial. This approach, called diversification, was proposed by Giesbrecht and Roche (2011). Our algorithm interpolates each variable independently by using
O
(
T
) probes, and then uses the diversification to correlate terms in different images. At last, we get the exponents by solving the discrete logarithms and obtain the coefficients by solving a linear system. We have implemented our algorithm in Maple. Experimental results show that our algorithm can be applied to the sparse polynomials with large degree within a few minutes. We also analyze the success rate of the algorithm. |
|---|---|
| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1674-7283 1869-1862 |
| DOI: | 10.1007/s11425-020-1791-5 |