## Spectral Properties of Boolean Functions, Graphs and Graph States (2005)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Riera05spectralproperties,

author = {Constanza Riera},

title = {Spectral Properties of Boolean Functions, Graphs and Graph States},

year = {2005}

}

### OpenURL

### Abstract

### Citations

6955 |
A Mathematical Theory of Communication
- Shannon
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Citation Context ...In this section, we give a summary of binary linear codes, one of the topics related to the thesis. The aim of coding theory is the reliable transmission of information across noisy channels. Shannon =-=[83]-=- was the first to formulate information theory mathematically. Ideally, one wishes to find codes which transmit quickly, contain many valid codewords and can correct or at least detect many errors. Ho... |

2064 |
The Theory of Error-Correcting Codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ... as far away as it can be from the affine functions [57], which is a desirable cryptographic design goal. It is an open problem to classify all bent Boolean functions, although many results are known =-=[32, 53, 21, 34]-=-. Functions with a flat WHT spectrum are crucial in the design of cryptosystems to avoid linear cryptanalysis. This type of attack was first devised by Matsui and Yamagishi [55] in an attack on FEAL, ... |

1915 | How to share a secret
- Shamir
- 1979
(Show Context)
Citation Context ...cations of the generalised bent criteria to classical cryptography, and, more generally, of the results contained in this thesis. For instance, there are more potential applications to Secret Sharing =-=[10, 84]-=-, HFE [70] and to the potential cryptanalysis of stream and block ciphers. • We have seen (lemma 7.3) that the pivot transform applied on bipartite graphs gives as a result a bipartite graph. On the o... |

931 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1994
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Citation Context ...the graph associated to the code. There has recently been a lot of interest in quantum information theory (QIT) and concretely in quantum computing, especially since the discovery of Shor’s algorithm =-=[85]-=-, which can factor an integer in polynomial time, and thus, in theory, break the RSA cryptographic system. For the time being, practical quantum computers have not yet been built, but the construction... |

658 |
Can Quantum-Mechanical Description of Physical Reality
- Einstein, Podolsky, et al.
- 1935
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Citation Context ...enoted by |φ〉 |ψ〉 , |φψ〉 or |φ, ψ〉. Given the coordinates of |φ〉 and |ψ〉, we can compute the coordinates of |φ〉 ⊗ |ψ〉 as the set of all pairwise products of coordinates of |φ〉 and |ψ〉. Definition 2.9 =-=[35]-=- A state |ρ〉 in V ⊗ W that can be written as |φ〉 ⊗ |ψ〉 is called a product state. If |ρ〉 is not a product state, it is entangled. 22 ,sDefinition 2.10 A pure quantum state is a state which can be desc... |

649 | A Handbook of Integer Sequences
- Sloane
- 1973
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Citation Context ...nd being 3 n . It has been computed in [27] that the number of LC-orbits for connected graphs for n = 1 to n = 12 are 1, 1, 1, 2, 4, 11, 26, 101, 440, 3132, 40457, and 1274068, respectively (see also =-=[45, 38, 47, 26, 87]-=-). 3.3.2 The LC-orbit Occurs Within {I, H, N} n One can verify that N � x and H � xz. Therefore one can replace x and xz with N and H, respectively, so the transform set, {I, xz, x} becomes {I, H, N}.... |

543 | Differential Cryptanalysis of DES-like Cryptosystems
- Biham, Shamir
- 1991
(Show Context)
Citation Context ...hers and cryptographic hash functions. The technique of differential cryptanalysis was first introduced by Murphy [59] in an attack on FEAL-4, but was later improved and perfected by Biham and Shamir =-=[11, 12]-=- who used it to attack DES. Differential cryptanalysis is basically a chosen plaintext attack (there are, however, extensions that would allow a known plaintext or even a ciphertext-only attack) and r... |

456 | Teleporting an unknown quantum state via dual classical and Einstein-PodolskyRosen channels
- Bennett, Brassard, et al.
- 1993
(Show Context)
Citation Context ...always detect eavesdropping. This is because measurements on the quantum carrier of information disturb it and so leave traces. Entanglement of qubits has also been used in quantum teleportation (see =-=[8, 18]-=- for the teleportation of a single qubit, or [79] for the teleportation of N qubits). As in classical communication, an important element in quantum information theory is the correction of errors. Whi... |

453 |
Linear Cryptanalysis Method for DES Cipher,” Eurocrypt
- Matsui
- 1993
(Show Context)
Citation Context ... WHT spectrum are crucial in the design of cryptosystems to avoid linear cryptanalysis. This type of attack was first devised by Matsui and Yamagishi [55] in an attack on FEAL, and extended by Matsui =-=[54]-=- to attack DES. It is a known plaintext/ciphertext attack, meaning that the attacker must be able to obtain encrypted ciphertexts for some set of plaintexts of his choosing. The idea as conceived by M... |

426 |
Safeguarding cryptographic keys
- Blakley
- 1979
(Show Context)
Citation Context ...cations of the generalised bent criteria to classical cryptography, and, more generally, of the results contained in this thesis. For instance, there are more potential applications to Secret Sharing =-=[10, 84]-=-, HFE [70] and to the potential cryptanalysis of stream and block ciphers. • We have seen (lemma 7.3) that the pivot transform applied on bipartite graphs gives as a result a bipartite graph. On the o... |

416 | Error-detecting and error-correcting codes - Hamming - 1950 |

248 | How to break md5 and other hash functions
- Wang, Yu
- 2005
(Show Context)
Citation Context ...rg and Knudsen [62] as well as Lai, Massey and Murphy [51]. Differential cryptanalysis has also been useful in attacking other cryptographic algorithms such as hash functions, as shown by Wang and Yu =-=[95]-=-. The analysis of spectra w.r.t. {I, H, N} n tells us more about p(x) than is provided by the spectrum w.r.t. the WHT; for instance, the analysis of the spectra w.r.t. {I, H} n is related to a probabi... |

236 | Quantum error correction via codes over GF (4
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- 1998
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Citation Context ...〉 is any quantum state. However, as proven by Peter Shor, the information of one qubit can be spread onto several physical qubits by using a Quantum Error-Correcting Code (QECC) (see definition 2.18) =-=[40, 19]-=-. Now, if noise or decoherence (interaction with the environment) corrupts one qubit, the information is not lost. The choice of I, H, and N, is motivated by their relevance to the set of equivalent Q... |

150 |
A single quantum cannot be cloned,” Nature 299
- Wooters, Zurek
- 1982
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Citation Context ...on of errors. While classical error-correction employs redundancy, this is not possible with quantum information, for, as stated by the no-cloning theorem, the information cannot be copied: Theorem 1 =-=[101]-=- There is no quantum operation that takes |φ〉 to |φ〉 ⊗ |φ〉, where |φ〉 is any quantum state. However, as proven by Peter Shor, the information of one qubit can be spread onto several physical qubits by... |

133 |
Synthesis of band-limited orthogonal signals for multichannel data transmission
- Chang
- 1966
(Show Context)
Citation Context ...[70] the functions used are quadratic. Recently, so-called generalised Boolean functions have been employed for wireless communication, concretely for OFDM (Orthogonal Frequency Division Multiplexing)=-=[22, 23]-=- systems. Recent applications of graph theory are for instance DNA sequencing by hybridization [3, 2], and the distributed networks in wireless communication. As we shall see, there is a close relatio... |

133 | Error correcting codes in quantum theory - Steane - 1996 |

128 | fields equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms
- Hidden
- 1996
(Show Context)
Citation Context ... generalised bent criteria to classical cryptography, and, more generally, of the results contained in this thesis. For instance, there are more potential applications to Secret Sharing [10, 84], HFE =-=[70]-=- and to the potential cryptanalysis of stream and block ciphers. • We have seen (lemma 7.3) that the pivot transform applied on bipartite graphs gives as a result a bipartite graph. On the other hand,... |

125 |
Nonlinearity Criteria for Cryptographic Functions
- Meier, Staffelbach
- 1989
(Show Context)
Citation Context ...se occurs when all Pk have equal magnitude, in which case p is said to have a flat WHT spectra, and is referred to as bent. If p is bent, then it is as far away as it can be from the affine functions =-=[57]-=-, which is a desirable cryptographic design goal. It is an open problem to classify all bent Boolean functions, although many results are known [32, 53, 21, 34]. Functions with a flat WHT spectrum are... |

115 | Markov ciphers and differential cryptanalysis
- Lai, Massey, et al.
- 1991
(Show Context)
Citation Context ... design of the S-boxes during the design of DES [24]. Studies on protecting ciphers against differential cryptanalysis have been conducted by Nyberg and Knudsen [62] as well as Lai, Massey and Murphy =-=[51]-=-. Differential cryptanalysis has also been useful in attacking other cryptographic algorithms such as hash functions, as shown by Wang and Yu [95]. The analysis of spectra w.r.t. {I, H, N} n tells us ... |

110 |
Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes
- Davis, Jedwab
- 1999
(Show Context)
Citation Context ... possible subset of {H, N} n transforms. The multivariate complementary set constructions of [68] provide candidate functions. The simplest and strongest of these is the line function (or path graph) =-=[81, 39, 31]-=-. 4.2.1 Line The line function in n variables, p l n(x), is defined as p l n−2 � n(x) = xjxj+1 + c · x + d , (4.1) j=0 where x, c ∈ GF(2) n , x = (x0, . . . , xn−1), and d ∈ GF(2). Its number of flat ... |

109 | Conjugate coding - Wiesner - 1983 |

104 |
Elementary Hadamard difference sets
- Dillon
- 1974
(Show Context)
Citation Context ... as far away as it can be from the affine functions [57], which is a desirable cryptographic design goal. It is an open problem to classify all bent Boolean functions, although many results are known =-=[32, 53, 21, 34]-=-. Functions with a flat WHT spectrum are crucial in the design of cryptosystems to avoid linear cryptanalysis. This type of attack was first devised by Matsui and Yamagishi [55] in an attack on FEAL, ... |

93 |
Complementary series
- Golay
- 1961
(Show Context)
Citation Context ... possible subset of {H, N} n transforms. The multivariate complementary set constructions of [68] provide candidate functions. The simplest and strongest of these is the line function (or path graph) =-=[81, 39, 31]-=-. 4.2.1 Line The line function in n variables, p l n(x), is defined as p l n−2 � n(x) = xjxj+1 + c · x + d , (4.1) j=0 where x, c ∈ GF(2) n , x = (x0, . . . , xn−1), and d ∈ GF(2). Its number of flat ... |

86 |
Generalized Hamming weights for linear codes
- Wei
- 1991
(Show Context)
Citation Context ...riable interlace polynomials by analogy, and investigate some of their properties. Next, for bipartite graphs, we develop a three-variable interlace polynomial and from it derive the weight hierarchy =-=[46, 99]-=- of the associated binary code, thereby expanding on the results of Parker and Rijmen [67]. Chapter 7 gives an expression of the pivot operation on graphs using the ANF of its associated (quadratic) B... |

85 | Scheme for reducing decoherence in quantum memory - Shor - 1995 |

72 |
Dirac The Principles of Quantum Mechanics
- M
- 1947
(Show Context)
Citation Context ...PU,k) = Us ∈ C 2n . (2.1) We offer here a brief introduction to some of the concepts in quantum theory, more concretely in quantum information theory, that are more relevant to this work. We refer to =-=[33, 93, 40]-=- for a more extended view of these subjects. Definition 2.7 [33] A quantum state |φ〉 is a unit vector 1 in a complex Hilbert space H. 1 It is enough to take a unit vector as representative for a state... |

69 |
Linear approximations of block ciphers
- Nyberg
- 1994
(Show Context)
Citation Context ... approximations might allow for a reduction in the amount of data required for a successful attack. Other issues such as protecting ciphers against linear cryptanalysis have been considered by Nyberg =-=[61]-=-, Knudsen [50], and O’Conner [63]. The autocorrelation of the function w.r.t. the WHT is used for differential cryptanalysis. This is a type of attack applicable primarily to iterative block ciphers, ... |

52 |
nauty User’s Guide
- McKay
(Show Context)
Citation Context ... of the classification of self-dual quantum codes, which is isomorphic to the classification of LC graph orbits, as described in [29, 30] and available at [26]. This classification in turn used nauty =-=[60]-=- to deal efficiently with graph isomorphism. The subsequent enumeration of pivot orbits of unlabelled connected graphs is shown in Table 7.1 up to n = 11. We have also classified and enumerated all pi... |

50 | C.: Algebraic Attacks and Decomposition of Boolean Functions
- Meier, Pasalic, et al.
- 2004
(Show Context)
Citation Context ...us more about p(x) than is provided by the spectrum w.r.t. the WHT; for instance, the analysis of the spectra w.r.t. {I, H} n is related to a probabilistic version of the so-called algebraic immunity =-=[56]-=-, as it identifies the linear or affine approximations to a Boolean function after fixing some of the variables (that is, taking xi = 0 or 1 for some i). More generally, {I, H, N} n can improve linear... |

48 |
Experimental quantum teleportation. Nature 390: 575–579
- Bouwmeester, Pan, et al.
- 1997
(Show Context)
Citation Context ...always detect eavesdropping. This is because measurements on the quantum carrier of information disturb it and so leave traces. Entanglement of qubits has also been used in quantum teleportation (see =-=[8, 18]-=- for the teleportation of a single qubit, or [79] for the teleportation of N qubits). As in classical communication, an important element in quantum information theory is the correction of errors. Whi... |

41 |
Quantum error-correcting codes associated with graphs. e-print quant–ph/0012111
- Schlingemann, Werner
(Show Context)
Citation Context ...substrate’ for measurementdriven quantum computation. Measurement-driven quantum computation on a quantum factor graph 2 has been discussed by Parker [66]. Independent work by Schlingemann and Werner =-=[82]-=-, Glynn [37, 38], and by Grassl, Klappenecker, and Rotteler [41] proposed to describe stabilizer Quantum 2 A factor graph is the diagram showing how a function of several variables can be factored int... |

38 | The interlace polynomial of a graph
- Arratia, Bollobás, et al.
(Show Context)
Citation Context ...work motivated, in part, by the application of interlace polynomials (see chapters 5 and 6) to the reconstruction of DNA strings [3, 2]. In particular, various interlace polynomials have been defined =-=[2, 1, 4, 5]-=- which mirror some of the quadratic results of chapter 4. 1.4 The Coding Theory Context In this section, we give a summary of binary linear codes, one of the topics related to the thesis. The aim of c... |

38 |
Differential-Linear Cryptanalysis
- Langford, Hellman
- 1994
(Show Context)
Citation Context ...cess. Attacks using linear cryptanalysis have been developed for block ciphers and stream ciphers. There have been a variety of enhancements and improvements to the basic attack. Langford and Hellman =-=[52]-=- introduced an attack called differential-linear cryptanalysis, combining elements of differential cryptanalysis (see below) with those of linear cryptanalysis. Also, Kaliski and Robshaw [48] showed t... |

35 |
Two new classes of bent functions
- Carlet
- 1994
(Show Context)
Citation Context ... as far away as it can be from the affine functions [57], which is a desirable cryptographic design goal. It is an open problem to classify all bent Boolean functions, although many results are known =-=[32, 53, 21, 34]-=-. Functions with a flat WHT spectrum are crucial in the design of cryptosystems to avoid linear cryptanalysis. This type of attack was first devised by Matsui and Yamagishi [55] in an attack on FEAL, ... |

34 | Provable security against a differential attack
- Nyberg, Knudsen
- 1995
(Show Context)
Citation Context ...ess is limited by what was very careful design of the S-boxes during the design of DES [24]. Studies on protecting ciphers against differential cryptanalysis have been conducted by Nyberg and Knudsen =-=[62]-=- as well as Lai, Massey and Murphy [51]. Differential cryptanalysis has also been useful in attacking other cryptographic algorithms such as hash functions, as shown by Wang and Yu [95]. The analysis ... |

33 |
A theoretical study of performance of an orthogonal multiplexing data transmission scheme
- Chang, Gibby
- 1968
(Show Context)
Citation Context ...[70] the functions used are quadratic. Recently, so-called generalised Boolean functions have been employed for wireless communication, concretely for OFDM (Orthogonal Frequency Division Multiplexing)=-=[22, 23]-=- systems. Recent applications of graph theory are for instance DNA sequencing by hybridization [3, 2], and the distributed networks in wireless communication. As we shall see, there is a close relatio... |

33 |
Some theorems on Fourier coefficients
- Rudin
- 1959
(Show Context)
Citation Context ... possible subset of {H, N} n transforms. The multivariate complementary set constructions of [68] provide candidate functions. The simplest and strongest of these is the line function (or path graph) =-=[81, 39, 31]-=-. 4.2.1 Line The line function in n variables, p l n(x), is defined as p l n−2 � n(x) = xjxj+1 + c · x + d , (4.1) j=0 where x, c ∈ GF(2) n , x = (x0, . . . , xn−1), and d ∈ GF(2). Its number of flat ... |

32 | Graphical description of the action of local Clifford transformations on graph states - Nest, Dehaene, et al. |

31 | Differential Cryptanalysis of Feal and NHash
- Biham, Shamir
- 1991
(Show Context)
Citation Context ...hers and cryptographic hash functions. The technique of differential cryptanalysis was first introduced by Murphy [59] in an attack on FEAL-4, but was later improved and perfected by Biham and Shamir =-=[11, 12]-=- who used it to attack DES. Differential cryptanalysis is basically a chosen plaintext attack (there are, however, extensions that would allow a known plaintext or even a ciphertext-only attack) and r... |

30 | The interlace polynomial: a new graph polynomial
- Arratia, Bollobas, et al.
- 1990
(Show Context)
Citation Context ...rpretations of the Interlace Polynomial The results for this chapter can be found in [76] and [78]. 5.1 Overview The (one-variable) interlace polynomial was introduced by Arratia, Bollobás and Sorkin =-=[2, 4]-=-, as a variant of Tutte and Tutte-Martin polynomials [17]. They defined the interlace polynomial of a graph G, q(G; z), by means of a recurrence formula, involving local complementation (LC) of the gr... |

28 |
Graphic presentations of isotropic systems
- Bouchet
- 1988
(Show Context)
Citation Context ...hole orbit of the graph under a graph operation. This is of significant relevance to the subject of this thesis (see chapters 3 and 5): 13sDefinition 5 Define the action of Local Complementation (LC) =-=[16, 15]-=- (or vertexneighbour-complement (VNC) [37]) on a graph G at vertex v, LCv, as the graph transformation obtained by replacing the subgraph defined by restricting the set of vertices to Nv, G[Nv], by it... |

27 | Practically Secure Feistel Ciphers
- Knudsen
- 1994
(Show Context)
Citation Context ...s might allow for a reduction in the amount of data required for a successful attack. Other issues such as protecting ciphers against linear cryptanalysis have been considered by Nyberg [61], Knudsen =-=[50]-=-, and O’Conner [63]. The autocorrelation of the function w.r.t. the WHT is used for differential cryptanalysis. This is a type of attack applicable primarily to iterative block ciphers, but also to st... |

27 |
A.: A new method for known plaintext attack of feal cipher
- Matsui, Yamagishi
- 1992
(Show Context)
Citation Context ... are known [32, 53, 21, 34]. Functions with a flat WHT spectrum are crucial in the design of cryptosystems to avoid linear cryptanalysis. This type of attack was first devised by Matsui and Yamagishi =-=[55]-=- in an attack on FEAL, and extended by Matsui [54] to attack DES. It is a known plaintext/ciphertext attack, meaning that the attacker must be able to obtain encrypted ciphertexts for some set of plai... |

26 |
Isotropic systems
- Bouchet
- 1987
(Show Context)
Citation Context ...Hein, Eisert and Briegel [45], and also by Van Den Nest and De Moor under the name of Local Clifford operation [94, 93]. VNC is another name for Local Complementation (LC), as investigated by Bouchet =-=[14, 15, 16]-=- in the context of isotropic systems. By applying LC to a graph G we obtain a graph G ′ , in which case we say that G and G ′ are LC-equivalent. Moreover, the set of all LC-equivalent graphs form an L... |

26 |
Uber extreme Punkt-und Kantenmengen
- Gallai
- 1959
(Show Context)
Citation Context ...oblem is strongly related to the independent set problem, because for any graph G := (V, E), the size of the smallest vertex covering plus the size of the maximum independent set equals the size of V =-=[36]-=-. Graphs have an important application in quantum mechanics, as a certain significant type of quantum state can be described by a simple graph (see chapter 10). The graphical description of certain pu... |

21 |
The Weight Distribution of Irreducible Cyclic Codes whit
- Helleseth, Kløve, et al.
- 1977
(Show Context)
Citation Context ...1 ∂ k! l! n ∂xk Q(G; x, y) ∂yl Let v = (v0, . . . , vn−1) ∈ GF(2) n . We define the support of v by supp v = {i : vi �= 0} . For a set X ⊆ GF(2) n , we define then X = ∪v∈Xsupp v. 101sDefinition 6.13 =-=[46, 99]-=- The weight hierarchy of an [n, k, d] binary linear code C is the sequence (d1, . . . , dk), where where ‘≤’ means here ‘subspace of’. dr = min{supp D : D ≤ C, dim D = r} , In section 6.2, we saw that... |

19 | Spectral orbits and peak-to-average power ratio of Boolean functions with respect to the {I
- Danielsen, Parker
- 2005
(Show Context)
Citation Context ...(2) is said to have a flat spectrum w.r.t. a transform T : C 2n → C 2n bent iff it has a flat spectra w.r.t. the transform WHT. iff |Pk| = 1 ∀ k ∈ GF(2) n . The function will be called Definition 2.6 =-=[29]-=- The Peak-to-Average Power Ratio (PAR) of a vector s ∈ C2n, with respect to a set of 2 n × 2 n unitary transforms T, is PART(s) = 2 n max U ∈ T k ∈ Z n 2 2.2 Quantum theory (|PU,k| 2 ), where PU = (PU... |

16 | A two-variable interlace polynomial
- Arratia, Bollobás, et al.
(Show Context)
Citation Context ...s invariant under LC. Proof: From definition 5.2 and chapter 3, one can show that Q is invariant w.r.t. {I, H, N} n . But, as seen in section 3.3.3, this set defines the LC operation. Definition 5.12 =-=[2, 5]-=- The action of pivot on a graph, G, at two connected vertices, u and v, (i.e. where G contains the edge uv), is given by LC(v)LC(u)LC(v) - that is the action of LC at vertex v, then vertex u, then ver... |

16 | G.: “Generalised bent criteria for Boolean functions (I
- Riera, Parker
- 2006
(Show Context)
Citation Context ...= ω 2 = ω + 1. A GF(4)-additive code of length n is an additive subgroup of GF(4) n . 27 4 3 1 2sChapter 3 Generalised Bent Criteria for Boolean Functions The results for this chapter can be found in =-=[74]-=-. 3.1 Overview The classification of bent quadratic (degree-two) Boolean functions is well-known [53], and is made easier because the bent criterion is an invariant of affine transformation of the inp... |

15 | The quantum entanglement of binary and bipolar sequences
- Parker, Rijmen
- 2002
(Show Context)
Citation Context ...s and r, where m : GF(2) n → GF(s), p : GF(2) n → GF(r), and x ∈ GF(2) n , such that the j th element of u, uj = m(j)α p(j) , where j ∈ GF(2) n and uj is interpreted as a complex number. Definition 1 =-=[67]-=- In the context above, when u is fully-factorised using the tensor product, then m and p are affine functions and we say that u represents a generalised affine function. We are trying to answer the qu... |