On the Derivative Imbalance and Ambiguity of Functions
Bibliographic record
Abstract
In 2007, Carlet and Ding introduced two parameters, denoted by $Nb_F$ and\n$NB_F$, quantifying respectively the balancedness of general functions $F$\nbetween finite Abelian groups and the (global) balancedness of their\nderivatives $D_a F(x)=F(x+a)-F(x)$, $a\\in G\\setminus\\{0\\}$ (providing an\nindicator of the nonlinearity of the functions). These authors studied the\nproperties and cryptographic significance of these two measures. They provided\nfor S-boxes inequalities relating the nonlinearity $\\mathcal{NL}(F)$ to $NB_F$,\nand obtained in particular an upper bound on the nonlinearity which unifies\nSidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the\nWorkshop WCC 2009 and in its postproceedings in 2011, a further study of these\nparameters was made; in particular, the first parameter was applied to the\nfunctions $F+L$ where $L$ is affine, providing more nonlinearity parameters.\n In 2010, motivated by the study of Costas arrays, two parameters called\nambiguity and deficiency were introduced by Panario \\emph{et al.} for\npermutations over finite Abelian groups to measure the injectivity and\nsurjectivity of the derivatives respectively. These authors also studied some\nfundamental properties and cryptographic significance of these two measures.\nFurther studies followed without that the second pair of parameters be compared\nto the first one.\n In the present paper, we observe that ambiguity is the same parameter as\n$NB_F$, up to additive and multiplicative constants (i.e. up to rescaling). We\nmake the necessary work of comparison and unification of the results on $NB_F$,\nrespectively on ambiguity, which have been obtained in the five papers devoted\nto these parameters. We generalize some known results to any Abelian groups and\nwe more importantly derive many new results on these parameters.\n
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".