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A function *F*:**F**_{2}^{n}**F**_{2}^{n} is *almost perfect nonlinear* (APN) if, for every *a**b* in **F**_{2}^{n}, the equation *F*(*x*)+*F*(*x*+*a*)=*b* has at most two solutions in *F*_{2}^{n}. When used as an S-box in a block cipher, it contributes optimally to the resistance to differential cryptanalysis. The function *F* is *almost bent* (AB) if the minimum Hamming distance between all its *component* functions *v**F*, *v*∈**F**_{2}^{n} _{2}^{n} ) and all affine Boolean functions on **F**_{2}^{n} takes the maximal value 2^{n-1}-2*n* odd only and contribute optimally to the resistance to the linear cryptanalysis. Every AB function is APN, and in the *n* odd case, any quadratic APN function is AB. The APN and AB properties are preserved by affine equivalence: *F**F*' if *F*'=*A*_{1}*F**A*_{2}, where *A*_{1},*A*_{2} are affine permutations. More generally, they are preserved by CCZ-equivalence, that is, affine equivalence of the graphs of *F*: {(*x*,*F*(*xv*)) | *x*∈ **F**_{2n}} and of *F*'. Until recently, the only known constructions of APN and AB functions were CCZ-equivalent to power functions *F*(*x*)=*x ^{d}* over finite fields (

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E91-A No.12 pp.3665-3678

- Publication Date
- 2008/12/01

- Publicized

- Online ISSN
- 1745-1337

- DOI
- 10.1093/ietfec/e91-a.12.3665

- Type of Manuscript
- Special Section INVITED PAPER (Special Section on Signal Design and its Applications in Communications)

- Category

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Claude CARLET, "On Almost Perfect Nonlinear Functions" in IEICE TRANSACTIONS on Fundamentals,
vol. E91-A, no. 12, pp. 3665-3678, December 2008, doi: 10.1093/ietfec/e91-a.12.3665.

Abstract: A function *F*:**F**_{2}^{n}**F**_{2}^{n} is *almost perfect nonlinear* (APN) if, for every *a**b* in **F**_{2}^{n}, the equation *F*(*x*)+*F*(*x*+*a*)=*b* has at most two solutions in *F*_{2}^{n}. When used as an S-box in a block cipher, it contributes optimally to the resistance to differential cryptanalysis. The function *F* is *almost bent* (AB) if the minimum Hamming distance between all its *component* functions *v**F*, *v*∈**F**_{2}^{n} _{2}^{n} ) and all affine Boolean functions on **F**_{2}^{n} takes the maximal value 2^{n-1}-2*n* odd only and contribute optimally to the resistance to the linear cryptanalysis. Every AB function is APN, and in the *n* odd case, any quadratic APN function is AB. The APN and AB properties are preserved by affine equivalence: *F**F*' if *F*'=*A*_{1}*F**A*_{2}, where *A*_{1},*A*_{2} are affine permutations. More generally, they are preserved by CCZ-equivalence, that is, affine equivalence of the graphs of *F*: {(*x*,*F*(*xv*)) | *x*∈ **F**_{2n}} and of *F*'. Until recently, the only known constructions of APN and AB functions were CCZ-equivalent to power functions *F*(*x*)=*x ^{d}* over finite fields (

URL: https://global.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e91-a.12.3665/_p

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@ARTICLE{e91-a_12_3665,

author={Claude CARLET, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={On Almost Perfect Nonlinear Functions},

year={2008},

volume={E91-A},

number={12},

pages={3665-3678},

abstract={A function *F*:**F**_{2}^{n}**F**_{2}^{n} is *almost perfect nonlinear* (APN) if, for every *a**b* in **F**_{2}^{n}, the equation *F*(*x*)+*F*(*x*+*a*)=*b* has at most two solutions in *F*_{2}^{n}. When used as an S-box in a block cipher, it contributes optimally to the resistance to differential cryptanalysis. The function *F* is *almost bent* (AB) if the minimum Hamming distance between all its *component* functions *v**F*, *v*∈**F**_{2}^{n} _{2}^{n} ) and all affine Boolean functions on **F**_{2}^{n} takes the maximal value 2^{n-1}-2*n* odd only and contribute optimally to the resistance to the linear cryptanalysis. Every AB function is APN, and in the *n* odd case, any quadratic APN function is AB. The APN and AB properties are preserved by affine equivalence: *F**F*' if *F*'=*A*_{1}*F**A*_{2}, where *A*_{1},*A*_{2} are affine permutations. More generally, they are preserved by CCZ-equivalence, that is, affine equivalence of the graphs of *F*: {(*x*,*F*(*xv*)) | *x*∈ **F**_{2n}} and of *F*'. Until recently, the only known constructions of APN and AB functions were CCZ-equivalent to power functions *F*(*x*)=*x ^{d}* over finite fields (

keywords={},

doi={10.1093/ietfec/e91-a.12.3665},

ISSN={1745-1337},

month={December},}

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TY - JOUR

TI - On Almost Perfect Nonlinear Functions

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 3665

EP - 3678

AU - Claude CARLET

PY - 2008

DO - 10.1093/ietfec/e91-a.12.3665

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E91-A

IS - 12

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - December 2008

AB - A function *F*:**F**_{2}^{n}**F**_{2}^{n} is *almost perfect nonlinear* (APN) if, for every *a**b* in **F**_{2}^{n}, the equation *F*(*x*)+*F*(*x*+*a*)=*b* has at most two solutions in *F*_{2}^{n}. When used as an S-box in a block cipher, it contributes optimally to the resistance to differential cryptanalysis. The function *F* is *almost bent* (AB) if the minimum Hamming distance between all its *component* functions *v**F*, *v*∈**F**_{2}^{n} _{2}^{n} ) and all affine Boolean functions on **F**_{2}^{n} takes the maximal value 2^{n-1}-2*n* odd only and contribute optimally to the resistance to the linear cryptanalysis. Every AB function is APN, and in the *n* odd case, any quadratic APN function is AB. The APN and AB properties are preserved by affine equivalence: *F**F*' if *F*'=*A*_{1}*F**A*_{2}, where *A*_{1},*A*_{2} are affine permutations. More generally, they are preserved by CCZ-equivalence, that is, affine equivalence of the graphs of *F*: {(*x*,*F*(*xv*)) | *x*∈ **F**_{2n}} and of *F*'. Until recently, the only known constructions of APN and AB functions were CCZ-equivalent to power functions *F*(*x*)=*x ^{d}* over finite fields (

ER -