Polynomials with two values

Gathen, Joachim von zur; Roche, J.R.

doi:10.1007/bf01215917

Cited by 46 publications

(66 citation statements)

References 4 publications

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“…For symmetric functions, the largest known separation remains a factor of 2. We know from von zur Gathen's and Roche's work on polynomials [10] and quantum lower bounds using polynomials [4] that for symmetric f : Q E (f ) ≥ n 2 −O(n 0.548 ), thus the largest known separation is either optimal or close to being optimal.…”

Section: Our Resultsmentioning

confidence: 99%

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Exact Quantum Query Complexity of $$\text {EXACT}_{k,l}^n$$

Ambainis

Iraids

Nagaj

2017

SOFSEM 2017: Theory and Practice of Computer Science

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In the decision tree model, one's task is to compute a boolean function f : {0, 1} n → {0, 1} on an input x ∈ {0, 1} n that is accessible via queries to a black box (the black box hides the bits xi). In the quantum case, classical queries and computation are replaced by unitary transformations. A quantum algorithm is exact if it always outputs the correct value of f (in contrast to the standard model of quantum algorithms where the algorithm is allowed to be incorrect with a small probability). The minimum number of queries for an exact quantum algorithm computing the function f is denoted by QE(f ).We consider the following n bit function with 0 ≤ k ≤ l ≤ n:i.e. we want to give the answer 1 only when exactly k or l of the bits xi are 1. We construct a quantum query algorithm for this function and give lower bounds for it, with lower bounds matching the complexity of the algorithm in some cases (and almost matching it in other cases):• For all k, l: max{n − k, l} − 1 ≤ QE(EXACT n k,l ) ≤ max{n − k, l} + 1.

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Section: Our Resultsmentioning

confidence: 99%

“…The lower bound follows trivially from von zur Gathen's and Roche's work [10] and the polynomial method [4]. For the upper bound, we now present an algorithm.…”

mentioning

confidence: 99%

Exact Quantum Query Complexity of $$\text {EXACT}_{k,l}^n$$

Ambainis

Iraids

Nagaj

2017

SOFSEM 2017: Theory and Practice of Computer Science

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“…We shall later see the connection between bounding the degree of functions that take values in {0, 1, 2} to proving the existence of a not too large S such thatf (S) = 0. We note that the result of [GR97] actually implies the following corollary. We say that a Boolean function f is balanced if Pr x [f (x) = 0] = 1/2.…”

Section: Related Workmentioning

confidence: 83%

“…In [GR97], following [NS94], von zur Gathen and Roche studied the question of giving a lower bound on the real degree of non-constant symmetric Boolean functions. In other words, the problem is proving that there is a large set S such thatf (S) = 0.…”

Section: Related Workmentioning

confidence: 99%

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On the Minimal Fourier Degree of Symmetric Boolean Functions

Shpilka

Tal

2011

2011 IEEE 26th Annual Conference on Computational Complexity

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In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f :525 is the largest gap between consecutive prime numbers in {1, . . . , m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [MOS04]. Namely, we show that the running time of their algorithm is at most n O(k 0.525 ) ·poly(n, 2 k , log(1/δ)) where n is the number of variables, k is the size of the junta (i.e. number of relevant variables) and δ is the error probability. In particular, for k ≥ log(n) 1/(1−0.525) ≈ log(n) 2.1 our analysis matches the lower bound 2 k (up to polynomial factors).Our bound on the degree greatly improves the previous result of Kolountzakis et al. [KLM + 09] who proved that |S| = O(k/ log k).

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On the Algebraic Immunity of Symmetric Boolean Functions

Braeken

Preneel

2005

Progress in Cryptology - INDOCRYPT 2005

111

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In this paper, we analyze the algebraic immunity of symmetric Boolean functions. The algebraic immunity is a property which measures the resistance against the algebraic attacks on symmetric ciphers. We identify a set of lowest degree annihilators for symmetric functions and propose an efficient algorithm for computing the algebraic immunity of a symmetric function. The existence of several symmetric functions with maximum algebraic immunity is proven. In this way, we have found a new class of functions which have good implementation properties and maximum algebraic immunity.

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Polynomials with two values

Cited by 46 publications

References 4 publications

Exact Quantum Query Complexity of $$\text {EXACT}_{k,l}^n$$

Exact Quantum Query Complexity of $$\text {EXACT}_{k,l}^n$$

On the Minimal Fourier Degree of Symmetric Boolean Functions

On the Algebraic Immunity of Symmetric Boolean Functions

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