Solving polynomial systems over non-fields and applications to modular polynomial factoring

Sayak Chakrabarti, Ashish Dwivedi, Nitin Saxena

Abstract: We study the problem of solving a system of m polynomials in n variables over the ring of integers modulo a prime-power pk<math><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math>. The problem over finite fields is well studied in varied parameter settings. For small characteristic p=2<math><mi is="true">p</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math>, Lokshtanov et al. (SODA'17) initiated the study, for degree d=2<math><mi is="true">d</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math> systems, to improve the exhaustive search complexity of O(2n)⋅poly(m,n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msup><mo stretchy="false" is="true">)</mo><mo is="true">⋅</mo><mtext mathvariant="sans-serif" is="true">poly</mtext><mo stretchy="false" is="true">(</mo><mi is="true">m</mi><mo is="true">,</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> to O(20.8765n)⋅poly(m,n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">0.8765</mn><mi is="true">n</mi></mrow></msup><mo stretchy="false" is="true">)</mo><mo is="true">⋅</mo><mtext mathvariant="sans-serif" is="true">poly</mtext><mo stretchy="false" is="true">(</mo><mi is="true">m</mi><mo is="true">,</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math>; which currently is improved to O(20.6943n)⋅poly(m,n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">0.6943</mn><mi is="true">n</mi></mrow></msup><mo stretchy="false" is="true">)</mo><mo is="true">⋅</mo><mtext mathvariant="sans-serif" is="true">poly</mtext><mo stretchy="false" is="true">(</mo><mi is="true">m</mi><mo is="true">,</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> in Dinur (SODA'21). For large p but constant n, Huang and Wong (FOCS'96) gave a randomized poly(d,m,log⁡p)<math><mtext mathvariant="sans-serif" is="true">poly</mtext><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo is="true">,</mo><mi is="true">m</mi><mo is="true">,</mo><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">p</mi><mo stretchy="false" is="true">)</mo></math> time algorithm. Note that for growing n, system-solving is known to be intractable even with p=2<math><mi is="true">p</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math> and degree d=2<math><mi is="true">d</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math>.We devise a randomized poly(d,m,log⁡p)<math><mtext mathvariant="sans-serif" is="true">poly</mtext><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo is="true">,</mo><mi is="true">m</mi><mo is="true">,</mo><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">p</mi><mo stretchy="false" is="true">)</mo></math>-time algorithm to find a root of a given system of m integral polynomials of degrees bounded by d, in n variables, modulo a prime power pk<math><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math>; when n+k<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">k</mi></math> is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic p) to system-solving over Galois rings (i.e., characteristic pk<math><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math>); when k>1<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">1</mn></math> is constant. The challenge here is to find a lift of singular Fp<math><msub is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">F</mi></mrow><mrow is="true"><mi is="true">p</mi></mrow></msub></math>-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.Our algorithm has applications to factoring univariate polynomials over Galois rings. Given f∈Z[x]<math><mi is="true">f</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">Z</mi><mo stretchy="false" is="true">[</mo><mi is="true">x</mi><mo stretchy="false" is="true">]</mo></math> and a prime-power pk<math><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math> (k≥2<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">2</mn></math>), finding factors of fmodpk<math><mi is="true">f</mi><mspace width="0.25em" is="true"></mspace><mrow is="true"><mi mathvariant="normal" is="true">mod</mi></mrow><mspace width="0.25em" is="true"></mspace><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math> has a curious state-of-the-art. It is solved for large k by p-adic factoring algorithms (von zur Gathen, Hartlieb, ISSAC'96); but unsolved for small k. In particular, no nontrivial factoring method is known for k≥5<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">5</mn></math> (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree-δ factors of f(x)modpk<math><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">x</mi><mo stretchy="false" is="true">)</mo><mspace width="0.25em" is="true"></mspace><mrow is="true"><mi mathvariant="normal" is="true">mod</mi></mrow><mspace width="0.25em" is="true"></mspace><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math> could be exponentially many, as soon as k≥2<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">2</mn></math>. We give the first randomized poly(deg⁡(f),log⁡p)<math><mo stretchy="false" is="true">(</mo><mi mathvariant="normal" is="true">deg</mi><mo is="true">⁡</mo><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo><mo is="true">,</mo><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">p</mi><mo stretchy="false" is="true">)</mo></math>-time algorithm to find a degree-δ factor of f(x)modpk<math><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">x</mi><mo stretchy="false" is="true">)</mo><mspace width="0.25em" is="true"></mspace><mrow is="true"><mi mathvariant="normal" is="true">mod</mi></mrow><mspace width="0.25em" is="true"></mspace><msup is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup></math>, when k+δ<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">δ</mi></math> is constant. Our method has potential application in algebraic coding theory. In particular, extending algebraic geometric and Reed-Solomon codes to Galois rings could enable new and improved bounds on their underlying efficiency parameters.