English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Odd degree Chebyshev polynomials over a ring of modulo $2^w$ have two kindsof period. One is an "orbital period". Odd degree Chebyshev polynomials arebijection over the ring. Therefore, when an odd degree Chebyshev polynomialiterate affecting a factor of the ring, we can observe an orbit over the ring.The "orbital period" is a period of the orbit. The other is a "degree period".It is observed when changing the degree of Chebyshev polynomials with a fixedargument of polynomials. Both kinds of period have not been completely studied.In this paper, we clarify completely both of them. The knowledge about themenables us to efficiently solve degree decision problem of Chebyshev polynomialover the ring, and so a key-exchange protocol with Chebyshev polynomial overthe ring is not secure. In addition, we generalize the discussion and show thata key-exchange protocol with more generalized permutation polynomials whichbelong to a certain class is not secure.

Three Theorems on odd degree Chebyshev polynomials and more generalized permutation polynomials over a ring of module $2^w$