Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2009
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/0903.2016 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910427328282624 |
|---|---|
| author | Hernando, Fernando McGuire, Gary |
| author_facet | Hernando, Fernando McGuire, Gary |
| contents | We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$. Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$ with two zeros $ω, ω^t$ has minimum distance 5 for infinitely many values of $n$. The conjecture we prove states that every exceptional number has the form $2^i+1$ or $4^i-2^i+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0903_2016 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions Hernando, Fernando McGuire, Gary Information Theory Algebraic Geometry We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$. Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$ with two zeros $ω, ω^t$ has minimum distance 5 for infinitely many values of $n$. The conjecture we prove states that every exceptional number has the form $2^i+1$ or $4^i-2^i+1$. |
| title | Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions |
| topic | Information Theory Algebraic Geometry |
| url | https://arxiv.org/abs/0903.2016 |