{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T22:02:26Z","timestamp":1788300146742,"version":"build-2803163510"},"reference-count":26,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2022,3,15]]},"abstract":"<jats:p> The security of cryptosystems based on Chebyshev recursive relation, [Formula: see text], relies on the difficulty to find the large degree [Formula: see text] of Chebyshev polynomial for given parameters. But this relation cannot be used to evaluate [Formula: see text], if [Formula: see text] is very large. We investigate three other methods: matrix-multiplication-based evaluation, halve-and-square evaluation, and root-extraction-based evaluation. We find they have the same complexity [Formula: see text] over finite field [Formula: see text]. The result shows that the hardness of some cryptosystems based on modular Chebyshev polynomials is almost equivalent to the general discrete logarithm. This partially answers the question why such chaos-based cryptosystems have rarely been put into practice. <\/jats:p>","DOI":"10.1142\/s0218127422500444","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T10:34:30Z","timestamp":1646994870000},"source":"Crossref","is-referenced-by-count":2,"title":["A Note on the Insecurity of Cryptosystems Based on Chebyshev Polynomials"],"prefix":"10.1142","volume":"32","author":[{"given":"Zhengjun","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, P. R. 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