Tamper Detection against Unitary Operators

Security of a storage device against a tampering adversary has been a well-studied topic in classical cryptography. Such models give black-box access to an adversary, and the aim is to protect the stored message or abort the protocol if there is any tampering.In this work, we extend the scope of the theory of tamper detection codes against an adversary with quantum capabilities. We consider encoding and decoding schemes that are used to encode a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-qubit quantum message <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">|</mml:mo><mml:mi>m</mml:mi><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math> to obtain an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit quantum codeword <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">|</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>&amp;#x03C8;</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math>. A quantum codeword <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">|</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>&amp;#x03C8;</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math> can be adversarially tampered via a unitary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> from some known tampering unitary family <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">U</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="sans-serif">A</mml:mi><mml:mi mathvariant="sans-serif">d</mml:mi><mml:mi mathvariant="sans-serif">v</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> (acting on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:msup></mml:math>).Firstly, we initiate the general study of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">quantum tamper detection codes</mml:mtext></mml:mrow></mml:math>, which detect if there is any tampering caused by the action of a unitary operator. In case there was no tampering, we would like to output the original message. We show that quantum tamper detection codes exist for any family of unitary operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">U</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="sans-serif">A</mml:mi><mml:mi mathvariant="sans-serif">d</mml:mi><mml:mi mathvariant="sans-serif">v</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math>, such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">|</mml:mo><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">U</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="sans-serif">A</mml:mi><mml:mi mathvariant="sans-serif">d</mml:mi><mml:mi mathvariant="sans-serif">v</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo fence="false" stretchy="false">|</mml:mo><mml:mo>&amp;#x003C;</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>&amp;#x03B1;</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math> for some constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B1;</mml:mi><mml:mo>&amp;#x2208;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>6</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>; provided that unitary operators are not too close to the identity operator. Quantum tamper detection codes that we construct can be considered to be quantum variants of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">classical tamper detection codes</mml:mtext></mml:mrow></mml:math> studied by Jafargholi and Wichs ['15], which are also known to exist under similar restrictions.Additionally, we show that when the message set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow></mml:math> is classical, such a construction can be realized as a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">non-malleable code</mml:mtext></mml:mrow></mml:math> against any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">U</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="sans-serif">A</mml:mi><mml:mi mathvariant="sans-serif">d</mml:mi><mml:mi mathvariant="sans-serif">v</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> of size up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>&amp;#x03B1;</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math>.