Multilinear Maps from Obfuscation

Abstract

Abstract We provide constructions of multilinear groups equipped with natural hard problems from indistinguishability obfuscation, homomorphic encryption, and NIZKs. This complements known results on the constructions of indistinguishability obfuscators from multilinear maps in the reverse direction. We provide two distinct, but closely related constructions and show that multilinear analogues of the $${\text {DDH}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>DDH</mml:mtext></mml:math> assumption hold for them. Our first construction is symmetric and comes with a $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math> -linear map $$\mathbf{e }: {{\mathbb {G}}}^\kappa \longrightarrow {\mathbb {G}}_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>e</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mi>κ</mml:mi></mml:msup><mml:mo>⟶</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mrow></mml:math> for prime-order groups $${\mathbb {G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> and $${\mathbb {G}}_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math> . To establish the hardness of the $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math> -linear $${\text {DDH}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>DDH</mml:mtext></mml:math> problem, we rely on the existence of a base group for which the $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math> -strong $${\text {DDH}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>DDH</mml:mtext></mml:math> assumption holds. Our second construction is for the asymmetric setting, where $$\mathbf{e }: {\mathbb {G}}_1 \times \cdots \times {\mathbb {G}}_{\kappa } \longrightarrow {\mathbb {G}}_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>e</mml:mi><mml:mo>:</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>×</mml:mo><mml:mo>⋯</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>κ</mml:mi></mml:msub><mml:mo>⟶</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:mrow></mml:math> for a collection of $$\kappa +1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> prime-order groups $${\mathbb {G}}_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math> and $${\mathbb {G}}_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math> , and relies only on the 1-strong $${\text {DDH}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>DDH</mml:mtext></mml:math> assumption in its base group. In both constructions, the linearity $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math> can be set to any arbitrary but a priori fixed polynomial value in the security parameter. We rely on a number of powerful tools in our constructions: probabilistic indistinguishability obfuscation, dual-mode NIZK proof systems (with perfect soundness, witness-indistinguishability, and zero knowledge), and additively homomorphic encryption for the group $$\mathbb {Z}_N^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>Z</mml:mi><mml:mi>N</mml:mi><mml:mo>+</mml:mo></mml:msubsup></mml:math> . At a high level, we enable “bootstrapping” multilinear assumptions from their simpler counterparts in standard cryptographic groups and show the equivalence of PIO and multilinear maps under the existence of the aforementioned primitives.