## Constructions of Bent Functions from Two Known Bent Functions (1994)

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Venue: | Australasian J. Comb |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Seberry94constructionsof,

author = {Jennifer Seberry and Xian-mo Zhang},

title = {Constructions of Bent Functions from Two Known Bent Functions},

journal = {Australasian J. Comb},

year = {1994},

volume = {9},

pages = {21--34}

}

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### Abstract

A (1, -1)-matrix will be called a bent type matrix if each row and each column are bent sequences. A similar description can be found in Carlisle M. Adams and Stafford E. Tavares, Generating and counting binary sequences, IEEE Trans. Inform. Theory, vol. 36, no. 5, pp. 1170-1173, 1990, in which the authors use the properties of bent type matrices to construct a class of bent functions. In this paper we give a general method to construct bent type matrices and show that the bent sequence obtained from a bent type matrix is a generalized result of the Kronecker product of two known bent sequences. Also using two known bent sequences of length 2 2k\Gamma2 we can construct 2 k \Gamma 2 bent sequences of length 2 2k , more than in the ordinary construction, which gives construct 10 bent sequences of length 2 2k from two known bent sequences of length length 2 2k\Gamma2 . Let V n be the vector space of n tuples of elements from GF (2). Let ff; fi 2 V n . Write ff = (a 1 ; \Delta ...

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Citation Context .... Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], [8], =-=[10]-=-, [9], [11], [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : : , a k\Gamma1 = b k\Gamma1... |

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On “bent” functions
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Citation Context ...fi;xi = \Sigma1; for every fi 2 V n . We call f(x) a bent function on V n . From Definition 2, bent functions on V n only exist for even n. Bent functions were first introduced and studied by Rothaus =-=[13]-=-. Further properties, constructions and equivalence bounds for bent functions can be found in [2], [5], [7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q ... |

117 |
Nonlinearity Criteria for Cryptographic Functions
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Citation Context ...r, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], [8], [10], =-=[9]-=-, [11], [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : : , a k\Gamma1 = b k\Gamma1 and ... |

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Citation Context ...holtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], [8], [10], [9], =-=[11]-=-, [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : : , a k\Gamma1 = b k\Gamma1 and a k = ... |

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Citation Context ...V n only exist for even n. Bent functions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in [2], [5], [7], =-=[12]-=-, [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7],... |

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Citation Context ...ions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in [2], [5], [7], [12], [16]. Kumar, Scholtz and Welch =-=[6]-=- defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], [8], [10], [9], [11], [12]. We say ... |

37 |
Generating and counting binary bent sequences
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Citation Context ...t functions on V n only exist for even n. Bent functions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in =-=[2]-=-, [5], [7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3]... |

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Citation Context ...ly exist for even n. Bent functions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in [2], [5], [7], [12], =-=[16]-=-. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], [8], ... |

18 | The use of bent sequences to achieve higher-order strict avalanche criterion
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Citation Context ...[2], [5], [7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography =-=[3]-=-, [1], [4], [7], [8], [10], [9], [11], [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : :... |

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Citation Context ...s on V n only exist for even n. Bent functions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in [2], [5], =-=[7]-=-, [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4]... |

17 |
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Citation Context ...[7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], =-=[4]-=-, [7], [8], [10], [9], [11], [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : : , a k\Gam... |

16 |
Bounds on the linear span of bent sequences
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Citation Context ...ctions on V n only exist for even n. Bent functions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in [2], =-=[5]-=-, [7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1]... |

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Citation Context ...[5], [7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], =-=[1]-=-, [4], [7], [8], [10], [9], [11], [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : : , a ... |

10 |
Combinatorics: Room Squares, sum-free sets
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(Show Context)
Citation Context ...ne function (a linear function). Definition 4 A (1, -1)-matrix H of order h will be called an Hadamard matrix if HH T = hI h . If h is the order of an Hadamard matrix then h is 1, 2 or divisible by 4 =-=[15]-=-. A special kind of Hadamard matrices defined as following will be relevant Definition 5 The Sylvester-Hadamard matrix ( or Walsh-Hadamard matrix) of order 2 n , denoted by H n , is generated by the r... |

8 |
Highly nonlinear 0-1 balanced functions satisfying strict avalanche criterion
- Seberry, Zhang
- 1993
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Citation Context ...pe Matrices Lemma 2 Write H n = 2 6 6 6 6 4 l 0 l 1 . . . l 2 n \Gamma1 3 7 7 7 7 5 where l i is a row of H n . Then l i is the sequence of a linear function on V n . Proof. The proof can be found in =-=[14]-=-. 2 We can now established Theorem 3 An (1, -1)-matrix of order 2 m \Theta 2 n is an affine type matrix if and only if each row is E n -constructed and each column is E m -constructed. Proof. Note tha... |

7 |
Decoding of sequences of bent functions by means of a fast Hadamard transform. Radiotechnika i elektronika
- Losev
- 1987
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Citation Context ... [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z n q to Z q . Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], =-=[8]-=-, [10], [9], [11], [12]. We say ff = (a 1 ; \Delta \Delta \Delta ; a n ) ! fi = (b 1 ; \Delta \Delta \Delta ; b n ) if there exists k, 1 ! = k ! = 2 n , such that a 1 = b 1 , : : : , a k\Gamma1 = b k\... |

7 |
On immunity against Biham and Shamir's \di erential cryptanalysis
- Adams
- 1992
(Show Context)
Citation Context ...2], [5], [7], [12], [16]. Kumar, Scholtz and Welch [6] de ned and studied the bent functions from Z n q to Zq. Bentfunctions are useful for digital communications, coding theory and cryptography [3], =-=[1]-=-, [4], [7], [8], [10], [9], [11], [12]. 1We say =(a 1� �an) < =(b 1� �bn) if there exists k, 1 < = k < = 2n , such that a 1 = b 1, :::, ak;1 = bk;1 and ak =0,bk = 1. Hence we can order all vectors in... |