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 Zap. Nauchn. Sem. POMI, 2008, Volume 358, Pages 54–76 (Mi znsl2145)

Time hierarchies for cryptographic function inversion with advice

E. A. Hirscha, D. Yu. Grigor'evb, K. V. Pervyshevcd

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Institute of Mathematical Research of Rennes
c Saint-Petersburg State University
d University of California, San Diego, Department of Computer Science and Engineering

Abstract: We prove a time hierarchy theorem for inverting functions computable in a slightly non-uniform polynomial time. In particular, we prove that if there is a strongly one-way function then for any $k$ and for any polynomial $p$, there is a function $f$ computable in linear time with one bit of advice such that there is a polynomial-time probabilistic adversary that inverts $f$ with probability $\ge1/p(n)$ on infinitely many lengths of input while all probabilistic $O(n^k)$-time adversaries with logarithmic advice invert $f$ with probability less than $1/p(n)$ on almost all lengths of input.
We also prove a similar theorem in the worst-case setting, i.e., if $\mathbf P\neq\mathbf{NP}$, then for every $l>k\ge1$
$$(\mathbf{DTime}[n^k]\cap\mathbf{NTime}[n])/1\subsetneq(\mathbf{DTime}[n^l]\cap\mathbf{NTime}[n])/1.$$
Bibl. – 16 titles.

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English version:
Journal of Mathematical Sciences (New York), 2009, 158:5, 633–644

UDC: 510.52

Citation: E. A. Hirsch, D. Yu. Grigor'ev, K. V. Pervyshev, “Time hierarchies for cryptographic function inversion with advice”, Studies in constructive mathematics and mathematical logic. Part XI, Zap. Nauchn. Sem. POMI, 358, POMI, St. Petersburg, 2008, 54–76; J. Math. Sci. (N. Y.), 158:5 (2009), 633–644

Citation in format AMSBIB
\Bibitem{HirGriPer08} \by E.~A.~Hirsch, D.~Yu.~Grigor'ev, K.~V.~Pervyshev \paper Time hierarchies for cryptographic function inversion with advice \inbook Studies in constructive mathematics and mathematical logic. Part~XI \serial Zap. Nauchn. Sem. POMI \yr 2008 \vol 358 \pages 54--76 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl2145} \transl \jour J. Math. Sci. (N. Y.) \yr 2009 \vol 158 \issue 5 \pages 633--644 \crossref{https://doi.org/10.1007/s10958-009-9403-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-67349115686}