Revisiting linearly extended discrete functions
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Bibliographic record
Abstract
Abstract The authors introduced a new family of cryptographic schemes in a previous research article, which includes many practical encryption schemes, such as the Feistel family. Given a finite field of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q , any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>></m:mo> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:math> n\gt m\ge 0 , the authors described a new way to extend discrete functions with domain size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> {q}^{m} and range size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> {q}^{n-m} to a permutation over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {q}^{n} elements using theory from linear error correcting codes. The authors previously showed that the knowledge about the differentials and correlations of the resulting permutation reduces solely to those of the extended discrete function. We show how the perfect secrecy of extended nonlinear functions transfers to the family of bijective linear extensions. We investigate how the concrete security of the family of nonlinear functions relates to the family of permutations obtained by such a type of linear extension. We also explore how the interplay between the entropy and the total variation distance (near-perfect secrecy with unbounded adversary) affects the mixing rate (number of iterations of the feedback linear extensions) with respect to the uniform distribution of the permutations over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {q}^{n} elements. We give a new proof that a distribution close to the uniform distribution has a large entropy.
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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 | 0.001 | 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 | 0.000 | 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 it