The four 2-ary Boolean functions with Hamming weight 1 are bent; i.e., their nonlinearity is 1 (these Hadamard matrices show the Hamming distance to each of the eight linear and affine functions). The following formula shows that a 2-ary function is bent when its nonlinearity is 1:
The Boolean function is bent; i.e., its nonlinearity is 6 (which is what these Hadamard Matrices show). The following formula shows that a 4-ary function is bent when its nonlinearity is 6:
In the mathematical field of combinatorics, a bent function is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables. Concretely, this means the maximum correlation between the output of the function and a linear function is minimal. In addition, the derivatives of a bent function are balanced Boolean functions, so for any change in the input variables there is a 50 percent chance that the output value will change.
The maximal nonlinearity means approximating a bent function by an affine (linear) function is hard, a useful property in the defence against linear cryptanalysis. In addition, detecting a change in the output of the function yields no information about what change occurred in the inputs, making the function immune to differential cryptanalysis.
Bent functions were defined and named in the 1960s by Oscar Rothaus in research not published until 1976.[1] They have been extensively studied for their applications in cryptography, but have also been applied to spread spectrum, coding theory, and combinatorial design. The definition can be extended in several ways, leading to different classes of generalized bent functions that share many of the useful properties of the original.
It is known that V. A. Eliseev and O. P. Stepchenkov studied bent functions, which they called minimal functions, in the USSR in 1962.[2] However, their results have still not been declassified.
Bent functions are also known as perfectly nonlinear (PN) boolean functions. Certain functions that are as close as possible to perfect nonlinearity (e.g. for functions of an odd number of bits, or vectorial functions) are known as almost perfectly nonlinear (APN).[3]
^Cite error: The named reference rothaus was invoked but never defined (see the help page).
^Cite error: The named reference bent-book was invoked but never defined (see the help page).
^Blondeau; Nyberg (2015-03-01). "Perfect nonlinear functions and cryptography". Finite Fields and Their Applications. 32: 120–147. doi:10.1016/j.ffa.2014.10.007. ISSN 1071-5797.
a bentfunction is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when...
number of zeros and ones. The Hamming weight of the function is the number of ones in the truth table. Bent: its derivatives are all balanced (the autocorrelation...
transformations Affine geometry 3D projection Homography Flat (geometry) Bentfunction Berger 1987, p. 38. Samuel 1988, p. 11. Snapper & Troyer 1989, p. 65...
The Bent Pyramid is an ancient Egyptian pyramid located at the royal necropolis of Dahshur, approximately 40 kilometres (25 mi) south of Cairo, built under...
subject to cryptanalysis such as the correlation attack. Bentfunction Balanced boolean functions that can be evaluated so that every input bit is unlikely...
Antilinear map – Conjugate homogeneous additive map Bentfunction – Special type of Boolean function Bounded operator – Linear transformation between topological...
input bits. Boolean functions which satisfy the highest order SAC are always bentfunctions, also called maximally nonlinear functions, also called "perfect...
with the exception of Bent Spear, Faded Giant and Dull Sword. AFI 10-206 notes that the flagword Pinnacle may be added to Bent Spear or Faded Giant to...
{\displaystyle x} . Find s {\displaystyle s} . Many functions, such as the Legendre symbol and bentfunctions, satisfy these constraints. With a quantum algorithm...
and algebraic coding theory, including work on self-dual codes and bentfunctions. Lee graduated from Ewha Womans University in 1971. After a master's...
ciphers • BEAR and LION ciphers • Beaufort cipher • Beaumanor Hall • Bentfunction • Berlekamp–Massey algorithm • Bernstein v. United States • BestCrypt...
Wales was the senior legal officer of the colony and functioned in many ways as a Chief Justice. Bent was the first barrister to be appointed to a judicial...
and CAST-256, whose S-boxes are based on the non-linear properties of bentfunctions. He also helped organize the first Selected Areas in Cryptography (SAC)...
bent during installation and for the long term. The former is somewhat larger than the latter. The minimum bend radius is in general also a function of...
A bent or indirect entrance is a defensive feature in medieval fortification. In a castle with a bent entrance, the gate passage is narrow and turns sharply...
a "bent normal" vector n ^ b {\displaystyle {\hat {n}}_{b}} is often generated, which points in the average direction of occluded samples. The bent normal...
together, Einar. Bent Fuglede died in Copenhagen on 7 December 2023, at the age of 98. Fuglede, Bent (1972). Finely Harmonic Functions. Lecture Notes in...
Kennedy December 7, 1964 (1964-12-07) An outsider befriends a woman who is bent on murdering her husband. 341 9 "Triumph" Harvey Hart Arthur A. Ross Ed Begley...
concept for a prototype touchscreen as well as a multi-function computer-configurable knob. Bent Stumpe was educated within the Royal Danish Air Force...
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied...
displacement. An example of a bent-axis in-line PTU can be found on the Hawker 4000 hydraulic system. In yet further representations, a bent-axis fixed-displacement...
Global Information
