https://rot256.dev/tags/lattices/Recent posts in Lattices on ROT256.en-usThu, 03 Sep 2026 00:00:00 +0000https://rot256.dev/post/dual-lattice/Thu, 03 Sep 2026 00:00:00 +0000https://rot256.dev/post/dual-lattice/<p>The dual lattice is:</p> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msup><mi>L</mi><mo>∗</mo></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">{</mo><mi mathvariant="bold">x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>∣</mo><mi mathvariant="normal">∀</mi><mi mathvariant="bold">v</mi><mo>∈</mo><mi>L</mi><mi mathvariant="normal">.</mi><mo stretchy="false">⟨</mo><mi mathvariant="bold">x</mi><mo separator="true">,</mo><mi mathvariant="bold">v</mi><mo stretchy="false">⟩</mo><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mo fence="true">}</mo></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr><mtr><mtd class ="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">{</mo><mi mathvariant="bold">x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>∣</mo><mi mathvariant="normal">∀</mi><mi mathvariant="bold">v</mi><mo>∈</mo><mi>L</mi><mi mathvariant="normal">.</mi><mo stretchy="false">⟨</mo><mi mathvariant="bold">x</mi><mo separator="true">,</mo><mi mathvariant="bold">v</mi><mo stretchy="false">⟩</mo><mo>≡</mo><mn>0</mn><mspace></mspace><mspace width="1em"/><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"/><mi mathvariant="double-struck">Z</mi><mo stretchy="false">)</mo><mo fence="true">}</mo></mrow></mrow></mstyle></mtd><mtd class ="mtr-glue"></mtd><mtd class ="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{align} \Ldual &amp;= \left\{ \vx \in \RR^n \mid \forall \vv \in \Lat. \ip{\vx}{\vv} \in \ZZ \right\} \\ &amp;= \left\{ \vx \in \RR^n \mid \forall \vv \in \Lat. \ip{\vx}{\vv} \equiv 0 \pmod{\ZZ} \right\} \end{align} </annotation></semantics></math></span><p>Spiritually, it serves a very similar role to the orthogonal complement for linear spaces:</p> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>S</mi><mo>⊥</mo></msup><mo>=</mo><mrow><mo fence="true">{</mo><mi mathvariant="bold">x</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>∣</mo><mi mathvariant="normal">∀</mi><mi mathvariant="bold">v</mi><mo>∈</mo><mi>S</mi><mi mathvariant="normal">.</mi><mo stretchy="false">⟨</mo><mi mathvariant="bold">x</mi><mo separator="true">,</mo><mi mathvariant="bold">v</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn><mo fence="true">}</mo></mrow></mrow><annotation encoding="application/x-tex"> \Sperp = \left\{ \vx \in \RR^n \mid \forall \vv \in \Sspace. \ip{\vx}{\vv} = 0 \right\} </annotation></semantics></math></span><p>The difference accounts for the &ldquo;discrete&rdquo; structure of lattices vs the continuous nature of linear subspaces:</p>