## Testing Shift-Equivalence of Polynomials Using Quantum Machines (1996)

Venue: | In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Grigoriev96testingshift-equivalence,

author = {D. Grigoriev},

title = {Testing Shift-Equivalence of Polynomials Using Quantum Machines},

booktitle = {In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation},

year = {1996},

pages = {49--54}

}

### OpenURL

### Abstract

1 Introduction In the paper we deal with the problem of testing, whether two given polynomials f; g 2 F [X1 ; : : : ; Xn ] are shift-equivalent, i.e. there exists a shift ff 1 ; : : : ; ff n such that f(X1+ff1 ; : : : ; Xn+ ff n) = g. The issue of considering polynomials up to the shifts appeared already in the context of the interpolation of shifted-sparse polynomials (see [8, 10, 7]), namely, the polynomials which become sparse after a suitable shift. We present the algorithms for computing the group Sf;f of the shifts (fi 1 ; : : : ; fi n) such that f(X1 + fi 1 ; : : : ; Xn + fi n) =<F10.66

### Citations

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Citation Context ...wer than a certain polynomial in n=d. For an arbitrary finite ground field F and the degree of f we design in the section 4 a quantum machine (for this computational model and the background see e.g. =-=[1, 13, 14, 16]-=-) which computes the group Sf;f . Observe that the developed in the section 4 methods actually allow one to design a quantum machine which for a given action of an abelian group on a finite set, compu... |

497 | Probability Theory
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Citation Context ...wer than a certain polynomial in n=d. For an arbitrary finite ground field F and the degree of f we design in the section 4 a quantum machine (for this computational model and the background see e.g. =-=[1, 13, 14, 16]-=-) which computes the group Sf;f . Observe that the developed in the section 4 methods actually allow one to design a quantum machine which for a given action of an abelian group on a finite set, compu... |

414 | Fast probabilistic algorithms for verification of polynomial identities
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(Show Context)
Citation Context ...ds for Sf;g also in the case under consideration. Let q = p m , a polynomial h 2 Fq [X1 ; : : : ; Xn ]. The following lemma 2 was told the author by R. Smolensky [15] and strengthens Schwartz's lemma =-=[12]-=- for finite fields. Observe that when nsq deg h and deg X i (h)sq \Gamma 2, 1sisn, lemma 2 follows from [9] (for arbitrary h a weaker bound was proved in [5]). Lemma 2 If h has a zero in F n q then h ... |

367 | On the Power of Quantum Computation
- Simon
- 1997
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Citation Context ... 2i a b p \Delta for a suitable b and f = f(X1+fi1 ; : : : ; Xn+fin) 2 Fq [X1 ; : : : ; Xn ] for some (fi 1 ; : : : ; fi n) 2 F n q , fi ` = X 1jm fi (j) ` w (j) 1s`sn, occurs with the amplitude (cf. =-=[13, 14, 18]-=-) 1 q n X ( P ff (j) 1 w (j) ;:::; P ff (j) n w (j) )2S f;f 1(ff (1) 1 + fi (1) 1 ) \Delta \Delta \Delta m(ff (m) 1 + fi (m) 1 ) \Delta \Delta \Delta nm\Gammam+1 (ff (1) n + fi (1) n ) \Delta \Delta \... |

286 | Quantum circuit complexity
- Yao
- 1993
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Citation Context ...wer than a certain polynomial in n=d. For an arbitrary finite ground field F and the degree of f we design in the section 4 a quantum machine (for this computational model and the background see e.g. =-=[1, 13, 14, 16]-=-) which computes the group Sf;f . Observe that the developed in the section 4 methods actually allow one to design a quantum machine which for a given action of an abelian group on a finite set, compu... |

150 | Quantum measurements and the Abelian stabilizer problem. Available as arXiv.org e-Print quant-ph/9511026 - Kitaev - 1995 |

90 |
Monte-carlo approximation algorithms for enumeration problems
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- 1989
(Show Context)
Citation Context ... : : : ; ff n such that f(X1+ff1 ; : : : ; Xn+ ff n) = g. The issue of considering polynomials up to the shifts appeared already in the context of the interpolation of shifted-sparse polynomials (see =-=[8, 10, 7]-=-), namely, the polynomials which become sparse after a suitable shift. We present the algorithms for computing the group Sf;f of the shifts (fi 1 ; : : : ; fi n) such that f(X1 + fi 1 ; : : : ; Xn + f... |

88 |
An approximate fourier transform useful in quantum factoring
- Coppersmith
- 1994
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Citation Context ...mpute in polynomial time the Fourier transform OE n for the cyclic group Zn of the order n for "smooth" n, namely n = p1 \Delta \Delta \Delta p ` being a product of pairwise distinct small p=-=rimes. In [4]-=- OE 2 k was computed by a quantum machine based on the fast Fourier transform. First we show (although we do not immediately use it below) that OE p k for any small p could be computed recursively on ... |

63 | Quantum cryptanalysis of hidden linear functions
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- 1995
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Citation Context ...up of a given element from the set (as the author recently learned, the problem of computing the stabilizator subgroup by a quantum machine was also solved in [17] with a better complexity bound). In =-=[18]-=- a quantum machine was constructed which allows one to test, whether a given function has a hidden linear structure, or to find the period of a periodic univariate function with small preimages (the l... |

39 |
Generalized polynomial remainder sequences
- Loos
- 1983
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Citation Context ...he size of the coefficients in a linear system representings" 1inS (i) can be bounded by n O(1) M ` , then the 2 size of the coefficients c j does not exceed (nd) O(1) M ` by the subresultant the=-=orem [11]-=-. Hence M `+1s(nd) O(1) M ` and we conclude that Mds(nd) O(d) M and the bit-size of all occurring coefficients is also less than M(nd) O(d) . Therefore, the running time of the described algorithm doe... |

31 | Verfahren der Schnellen Fouriertransformation [Fast Fourier Transform Methods - Beth - 1984 |

14 | A zero-test and an interpolation algorithm for the shifted sparse polynomials. In Applied algebra, algebraic algorithms and error-correcting codes - Grigoriev, Karpinski - 1993 |

12 | Algorithms for computing sparse shifts for multivariate polynomials
- Grigoriev, Lakshman
- 1995
(Show Context)
Citation Context ... : : : ; ff n such that f(X1+ff1 ; : : : ; Xn+ ff n) = g. The issue of considering polynomials up to the shifts appeared already in the context of the interpolation of shifted-sparse polynomials (see =-=[8, 10, 7]-=-), namely, the polynomials which become sparse after a suitable shift. We present the algorithms for computing the group Sf;f of the shifts (fi 1 ; : : : ; fi n) such that f(X1 + fi 1 ; : : : ; Xn + f... |

9 | An approximation algorithm for the number of zeros of arbitrary polynomials over GF[q
- Grigoriev, Karpinski
- 1991
(Show Context)
Citation Context ...sky [15] and strengthens Schwartz's lemma [12] for finite fields. Observe that when nsq deg h and deg X i (h)sq \Gamma 2, 1sisn, lemma 2 follows from [9] (for arbitrary h a weaker bound was proved in =-=[5]-=-). Lemma 2 If h has a zero in F n q then h has at least q n\Gammadeg(h) zeroes. Now we describe a randomized algorithm which computessSf;g ae F n p . Similar to the section 2 the algorithm by recursio... |

8 | Sparse shifts for univariate polynomials
- Lakshman, Saunders
- 1996
(Show Context)
Citation Context ... : : : ; ff n such that f(X1+ff1 ; : : : ; Xn+ ff n) = g. The issue of considering polynomials up to the shifts appeared already in the context of the interpolation of shifted-sparse polynomials (see =-=[8, 10, 7]-=-), namely, the polynomials which become sparse after a suitable shift. We present the algorithms for computing the group Sf;f of the shifts (fi 1 ; : : : ; fi n) such that f(X1 + fi 1 ; : : : ; Xn + f... |

2 |
Solving algebraic systems in subexponential time
- Chistov, Grigoriev
- 1983
(Show Context)
Citation Context ...e profitable for computing Sf;g to solve a system of polynomial equations f(X1 + ff 1 ; : : : ; Xn + ff n) = g(X1 ; : : : ; Xn) in n variables ff 1 ; : : : ; ff n with the running time (Md n 2 ) O(1) =-=[3]-=-. 3 Testing shift-equivalence of reduced polynomials over a prime residues field: randomized algorithm Let the polynomials f; g 2 Fp [X1 ; : : : ; Xn ], deg(f);deg(g)sd, where p is a prime, be reduced... |

2 | Efficient approximation algorithms for sparse polynomials over finite fields
- Karpinski, Shparlinski
- 1994
(Show Context)
Citation Context ...following lemma 2 was told the author by R. Smolensky [15] and strengthens Schwartz's lemma [12] for finite fields. Observe that when nsq deg h and deg X i (h)sq \Gamma 2, 1sisn, lemma 2 follows from =-=[9]-=- (for arbitrary h a weaker bound was proved in [5]). Lemma 2 If h has a zero in F n q then h has at least q n\Gammadeg(h) zeroes. Now we describe a randomized algorithm which computessSf;g ae F n p . ... |

2 |
Private communication
- Smolensky
- 1995
(Show Context)
Citation Context ...educed, lemma 1 from the section 2 holds for Sf;g also in the case under consideration. Let q = p m , a polynomial h 2 Fq [X1 ; : : : ; Xn ]. The following lemma 2 was told the author by R. Smolensky =-=[15]-=- and strengthens Schwartz's lemma [12] for finite fields. Observe that when nsq deg h and deg X i (h)sq \Gamma 2, 1sisn, lemma 2 follows from [9] (for arbitrary h a weaker bound was proved in [5]). Le... |