作者 Tokareva, Natalia, author.

標題 Bent functions : results and applications to cryptography / by Natalia Tokareva (Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia)

目錄
	Foreword					xi
	Preface					xiii
	Notation					xvii
1.	Boolean Functions					1
	Introduction					1
1.1.	Definitions					1
1.2.	Algebraic normal form					3
1.3.	Boolean cube and Hamming distance					4
1.4.	Extended affinely equivalent functions					6
1.5.	Walsh-Hadamard transform					7
1.6.	Finite field and boolean functions					8
1.7.	Trace function					9
1.8.	Polynomial representation of a boolean function					11
1.9.	Trace representation of a boolean function					11
1.10.	Monomial boolean functions					14
2.	Bent Functions: An Introduction					17
	Introduction					17
2.1.	Definition of a nonlinearity					17
2.2.	Nonlinearity of a random boolean function					18
2.3.	Definition of a bent function					18
2.4.	If n is odd?					20
2.5.	Open problems					21
2.6.	Surveys					23
3.	History of Bent Functions					25
	Introduction					25
3.1.	Oscar Rothaus					25
3.2.	V.A. Eliseev and O.P. Stepchenkov					26
3.3.	From the 1970s to the present					28
4.	Applications of Bent Functions					31
	Introduction					31
4.1.	Cryptography: linear cryptanalysis and boolean functions					31
4.2.	Cryptography: one historical example					32
4.3.	Cryptography: bent functions in CAST					34
4.4.	Cryptography: bent functions in Grain					35
4.5.	Cryptography: bent functions in HAVAL					36
4.6.	Hadamard matrices and graphs					37
4.7.	Links to coding theory					38
4.8.	Bent sequences					39
4.9.	Mobile networks, CDMA					40
4.10.	Remarks					42
5.	Properties of Bent Functions					43
	Introduction					43
5.1.	Degree of a bent function					43
5.2.	Affine transformations of bent functions					44
5.3.	Rank of a bent function					45
5.4.	Dual bent functions					45
5.5.	Other properties					46
6.	Equivalent Representations of Bent Functions					49
	Introduction					49
6.1.	Hadamard matrices					49
6.2.	Difference sets					49
6.3.	Designs					50
6.4.	Linear spreads					50
6.5.	Sets of subspaces					51
6.6.	Strongly regular graphs					52
6.7.	Bent rectangles					52
7.	Bent Functions with a Small Number of Variables					55
	Introduction					55
7.1.	Two and four variables					55
7.2.	Six variables					56
7.3.	Eight variables					59
7.4.	Ten and more variables					60
7.5.	Algorithms for generation of bent functions					61
7.6.	Concluding remarks					62
8.	Combinatorial Constructions of Bent Functions					63
	Introduction					63
8.1.	Rothaus's iterative construction					63
8.2.	Maiorana-McFarland class					64
8.3.	Partial spreads: PS+, PS--					65
8.4.	Dillon's bent functions: PSap					66
8.5.	Dobbertin's construction					67
8.6.	More iterative constructions					67
8.7.	Minterm iterative constructions					68
8.8.	Bent iterative functions: BI					69
8.9.	Other constructions					72
9.	Algebraic Constructions of Bent Functions					73
	Introduction					73
9.1.	An algebraic approach					73
9.2.	Bent exponents: general properties					74
9.3.	Gold bent functions					75
9.4.	Dillon exponent					76
9.5.	Kasami bent functions					76
9.6.	Canteaut-Leander bent functions (MF-1)					78
9.7.	Canteaut-Charpin-Kuyreghyan bent functions (MF-2)					78
9.8.	Niho exponents					79
9.9.	General algebraic approach					80
9.10.	Other constructions					80
10.	Bent Functions and Other Cryptographic Properties					81
	Introduction					81
10.1.	Cryptographic criteria					81
10.2.	High degree and balancedness					82
10.3.	Correlation immunity and resiliency					82
10.4.	Algebraic immunity					83
10.5.	Vectorial bent functions, almost bent functions, and almost perfect nonlinear functions					85
11.	Distances Between Bent Functions					89
	Introduction					89
11.1.	The minimal possible distance between bent functions					89
11.2.	Classification of bent functions at the minimal distance from the quadratic bent function					90
11.3.	Upper bound for the number of bent functions at the minimal distance from an arbitrary bent function					93
11.4.	Bent functions at the minimal distance from a McFarland bent function					94
11.5.	Locally metrically equivalent bent functions					94
11.6.	The graph of minimal distances of bent functions					95
12.	Automorphisms of the Set of Bent Functions					97
	Introduction					97
12.1.	Preliminaries					97
12.2.	Shifts of the class of bent functions					98
12.3.	Duality between definitions of bent and affine functions					102
12.4.	Automorphisms of the set of bent functions					104
12.5.	Metrically regular sets					105
13.	Bounds on the Number of Bent Functions					107
	Introduction					107
13.1.	Preliminaries					107
13.2.	The number of bent functions for small n					108
13.3.	Upper bounds					108
13.4.	Direct lower bounds					111
13.5.	Iterative lower bounds					112
13.6.	Lower bound from the bent iterative functions					114
13.7.	Testing of the lower bound for small n					118
13.8.	Asymptotic problem and hypotheses					120
14.	Bent Decomposition Problem					123
	Introduction					123
14.1.	Preliminaries					123
14.2.	Partial results					124
14.3.	Boolean function as the sum of a constant number of bent functions					125
14.4.	Any cubic boolean function in eight variables is the sum of at most four bent functions					127
14.5.	Decomposition of dual bent functions					128
15.	Algebraic Generalizations of Bent Functions					133
	Introduction					133
15.1.	Preliminaries					133
15.2.	The q-valued bent functions					134
15.3.	The p-ary bent functions					137
15.4.	Bent functions over a finite field					139
15.5.	Bent functions over quasi-frobenius local rings					141
15.6.	Generalized boolean bent functions (of Schmidt)					141
15.7.	Bent functions from a finite abelian group into the set of complex numbers on the unit circle					144
15.8.	Bent functions from a finite abelian group into a finite abelian group					145
15.9.	Non-abelian bent functions					147
15.10.	Vectorial G-bent functions					148
15.11.	Multidimensional bent functions on a finite abelian group					149
16.	Combinatorial Generalizations of Bent Functions					151
	Introduction					151
16.1.	Symmetric bent functions					151
16.2.	Homogeneous bent functions					152
16.3.	Rotation-symmetric bent functions					152
16.4.	Normal bent functions					155
16.5.	Self-dual and anti-self-dual bent functions					156
16.6.	Partially defined bent functions					158
16.7.	Plateaued functions					158
16.8.	Z-bent functions					159
16.9.	Negabent functions, bent4-functions, and l-bent functions					160
17.	Cryptographic Generalizations of Bent Functions					163
	Introduction					163
17.1.	Semibent functions (near-bent functions)					163
17.2.	Balanced (semi-) bent functions					165
17.3.	Partially bent functions					166
17.4.	Hyperbent functions					168
17.5.	Bent functions of higher order					171
17.6.	k-bent functions					172
	References					175
	Index					197