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Generalized hyper-bent functions over GF(p)

A.M. Youssef
2007 Discrete Applied Mathematics  
In this paper, we extend the concept of binary hyper-bent functions introduced by Carlet to functions defined over GF(p). We show that such functions must be quadratic.  ...  We also provide the necessary and sufficient conditions on the symmetric coefficient matrix corresponding to the quadratic form of f : Z n p → Z p that guarantee that f is a hyper-bent function.  ...  Results Here, we generalize the concept of hyper-bent functions to functions defined over GF(p). Definition 5.  ... 
doi:10.1016/j.dam.2006.11.007 fatcat:u6xnmvopgbfndijaxxm6zeesa4

Generalization of Boolean Functions Properties to Functions Defined over GF(p)

2012 Journal of Computer Science  
Approach: We generalize some of the previous results on cryptographic Boolean functions to functions defined over GF(p); p > 2.  ...  We also derive an upper bound for the dimension of the linear space of the functions defined over GF(p). Finally, we present a method to construct a bent function from semi-bent functions.  ...  The concept of hyper-bent function was extended to functions over GF(p) in (Youssef, 2007) .  ... 
doi:10.3844/jcssp.2012.809.814 fatcat:3bduurfp2bgtpjyghyrl5xyy4m

Hyper-bent Functions [chapter]

Amr M. Youssef, Guang Gong
2001 Lecture Notes in Computer Science  
In this paper we introduce a new class of bent functions which we call hyper-bent functions.  ...  We also extend our results to vectorial hyper-bent functions.  ...  We would like to thank Claude Carlet for encouraging us to keep the "hyper-bent" term and for attracting our attention to references [6] and [7] .  ... 
doi:10.1007/3-540-44987-6_25 fatcat:h57djycjm5dvxdvcw7bipowhdm

New designs for signal sets with low cross correlation, balance property, and large linear span: GF(p) case

Guang Gong
2002 IEEE Transactions on Information Theory  
New designs for families of sequences over GF ( ) with low cross correlation, balance property, and large linear span are presented.  ...  The key idea of the new designs is to use short -ary sequences of period with the two-level autocorrelation function together with the interleaved structure to construct a set of long sequences with the  ...  ACKNOWLEDGMENT The author wishes to thank P.  ... 
doi:10.1109/tit.2002.804044 fatcat:ozves4s3irhz5avtgfxedfefji

Boolean Functions with Large Distance to All Bijective Monomials: N Odd Case [chapter]

Amr Youssef, Guang Gong
2001 Lecture Notes in Computer Science  
Hyper-bent functions achieve the maximal minimum distance to all the coordinate functions of all bijective monomials.  ...  However, this class of functions exists only for functions with even number of inputs.  ...  At Eurocrypt' 2001, Youssef and Gong [19] introduced a new class of bent functions which they called hyper-bent functions.  ... 
doi:10.1007/3-540-45537-x_4 fatcat:ts7x4rvgazhxjlnokn5cqyqyx4

Page 5528 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
A 40 (1985), no. 1, 90-107; MR 871:05075}, bent functions over GF( p* ) defined by A. S. Ambrosimov [Diskret. Mat. 6 (1994), no. 3, 50-60; MR 96e:03033] and perfect nonlinear functions.  ...  Summary: “In this paper, the author generalizes perfect nonlin- ear functions and discusses the relationships among bent functions over Z,, defined by P. V. Kumar, R. A. Scholtz and L. R. Welch [J.  ... 

Bent functions with stronger nonlinear properties: K-bent functions

N. N. Tokareva
2008 Journal of Applied and Industrial Mathematics  
We introduce the notion of k-bent function, i.e., a Boolean function with even numberm of variables v 1 , . . . , v m that can be approximated with all functions of the form u, v j ⊕ a in the equally bad  ...  Namely, the vectors of values for the functions u, v k ⊕ a are codewords of the code A k m .  ...  For every even m, they proved the existence of such hyper-bent functions. C. Carlet and P. Gaborit [9] and independently A. S. Kuzmin, V. T. Markov, A. A. Nechaev, and A. B.  ... 
doi:10.1134/s1990478908040133 fatcat:wfszouacajbizk4r4h2eh27hii

On Relationships among Avalanche, Nonlinearity, and Correlation Immunity [chapter]

Yuliang Zheng, Xian-Mo Zhang
2000 Lecture Notes in Computer Science  
2 n−1− 1 2 p .  ...  We establish, for the first time, an explicit and simple lower bound on the nonlinearity N f of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, N f ≥ 2 n−1 −  ...  Zheng and Zhang [26] proved that Boolean functions satisfying the avalanche criterion in a hyper-space coincide with certain bent functions.  ... 
doi:10.1007/3-540-44448-3_36 fatcat:tuclmwzbqrapbjmm4e4vtiauxq

Connections among nonlinearity, avalanche and correlation immunity

Yuliang Zheng, Xian-Mo Zhang
2003 Theoretical Computer Science  
Over the years researchers have identiÿed a number of indicators that forecast nonlinear properties of these functions.  ...  2 n−1−(1=2)p .  ...  Zheng and Zhang [29] proved that Boolean functions satisfying the avalanche criterion in a hyper-space coincide with certain bent functions.  ... 
doi:10.1016/s0304-3975(02)00319-5 fatcat:4goegm2qojcyrj7xke4oopdsyy

On Pivot Orbits of Boolean Functions [article]

Constanza Riera, Lars Eirik Danielsen, Matthew G. Parker
2006 arXiv   pre-print
We establish lower bounds on the number of flat spectra of a Boolean function, depending on internal structures, with respect to the I,H^n and I,H,N^n sets of transforms.  ...  We also construct a family of Boolean functions of degree higher than two with a large number of flat spectra with respect to I,H^n, and compute a lower bound on this number.  ...  The example used in the proof of lemma 3 was taken from an interesting catalogue of homogeneous bent functions, as provided by [5] . Theorem 7. Let m : GF(2) n → GF(2) and p : GF(2) n → Z 4 .  ... 
arXiv:math/0604396v1 fatcat:lxztvjpc4feydftzfwgwzu24d4

On Cryptographic Propagation Criteria for Boolean Functions

Claude Carlet
1999 Information and Computation  
We finally give a general construction of nonquadratic functions satisfying EPC(l) of order k.  ...  We determine the functions on GF(2) n which satisfy the propagation criterion of degree n&2, PC(n&2).  ...  For n even, these functions were characterized in [8] . They were called hyper-bent. We give this characterization in Proposition 7 for n even and extend it to n odd.  ... 
doi:10.1006/inco.1998.2763 fatcat:yjehp6r6prc5znycgb56trc5a4

On O(a^2) effects in gradient flow observables [article]

Alberto Ramos, Stefan Sint
2015 arXiv   pre-print
function Σ(u, a/L) =ḡ 2 GF (2L) at u =ḡ 2 GF (L) = 2.6057 for lattice sizes L/a = 8, 12, 16, 24 and with scale factor s = 2.  ...  We here follow the convention where c 2 multiplies the "bent rectangles" or "chairs", and c 3 the "parallelograms". where corrections are O(a 4 ) and J n,m = t (m+n)/2 ∞ −∞ d 4 p e −2t p 2 p n p m ∞ −∞  ... 
arXiv:1411.6706v2 fatcat:igbz4syzu5bibpwgc2rmfoeg7e

On the propagation criterion of degree l and order k [chapter]

Claude Carlet
1998 Lecture Notes in Computer Science  
We determine those Boolean functions on GF(2)" which satisfy the propagation criterion of degree l and order k _> n -/ -2. All of theses functions are quadratic.  ...  We design nonquadratic Boolean functions satisfying the criterion PC(e) of order k by using the Maiorana-McFarland construction involving nonlinear mappings derived from nonlinear codes.  ...  Preparata code) being greater than or equal to 2 'r~-I -2"~ (resp. being equal to 3) , the function f of corollary 18 can be defined so that it is (2 'n-1 -2~ -1)resilient (resp. 2-resilient).  ... 
doi:10.1007/bfb0054146 fatcat:iqg5k6ababd5fhvl6xv6z7ox3i

On Exact Algebraic [Non-]Immunity of S-Boxes Based on Power Functions [chapter]

Nicolas T. Courtois, Blandine Debraize, Eric Garrido
2006 Lecture Notes in Computer Science  
These attacks are not always feasible in practice but are in general very powerful.  ...  In this paper we are interested in algebraic immunity of several well known highly-nonlinear vectorial Boolean functions (or Sboxes), designed for block and stream ciphers.  ...  Gong: Hyper-bent Functions, In Eurocrypt 2001, LNCS 2045, pp 406-419, Springer, 2001.  ... 
doi:10.1007/11780656_7 fatcat:aiqmjpgmcrawdkjjie4n745scq

Effect of renal function on renal responsiveness to parathyroid hormone in primary hyperparathyroidism and chronic renal failure

1985 European Journal of Clinical Investigation  
Urinary cyclic AMP analysed as a function of the serum calcium and parathyroid hormone in the differential diagnosis of hyper- calcaemia. J Clin Invest 1977; 59:14-21.  ...  Like Schneider & Jutzler [5], we found a close correlation between basal plasma cAMP and plasma creatinine over a wide range of impaired renal function. High basal plasma cAMP concentra- 42 I. G.  ... 
doi:10.1111/j.1365-2362.1985.tb00141.x pmid:2985400 fatcat:qtykerkyynfxnovbqfsi2xezuy
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