{"paper":{"title":"Differential uniformity and second order derivatives for generic polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Fabien Herbaut (IMATH), I2M), Yves Aubry (IMATH","submitted_at":"2017-03-21T16:20:52Z","abstract_excerpt":"For any polynomial  $f$ of ${\\mathbb F}\\_{2^n}[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$$\\delta^2(f):=\\max\\_{\\substack{\\alpha \\in {\\mathbb F}\\_{2^n}^{\\ast} ,\\alpha' \\in {\\mathbb F}\\_{2^n}^{\\ast} ,\\beta \\in {\\mathbb F}\\_{2^n} \\alpha\\not=\\alpha'}} \\sharp\\{x\\in{\\mathbb F}\\_{2^n} \\mid D\\_{\\alpha,\\alpha'}^2f(x)=\\beta\\}$$where $D\\_{\\alpha,\\alpha'}^2f(x):=D\\_{\\alpha'}(D\\_{\\alpha}f(x))=f(x)+f(x+\\alpha)+f(x+\\alpha')+f(x+\\alpha+\\alpha')$ is  the second order derivative.Our purpose is to prove a dens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.07299","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}