International Association
for Cryptologic Research

We provide the first attribute based encryption (ABE) scheme for Turing machines supporting unbounded collusions from lattice assumptions. In more detail, the encryptor encodes an attribute $\vecx$ together with a bound $t$ on the machine running time and a message $m$ into the ciphertext, the key generator embeds a Turing machine $M$ into the secret key and decryption returns $m$ if and only if $M(\vecx)=1$. Crucially, the input $\vecx$ and machine $M$ can be of unbounded size, the time bound $t$ can be chosen dynamically for each input and decryption runs in input specific time. \smallskip Previously the best known ABE for uniform computation supported only non-deterministic log space Turing machines (${\sf NL})$ from pairings (Lin and Luo, Eurocrypt 2020). In the post-quantum regime, the state of the art supports non-deterministic finite automata from LWE in the {\it symmetric} key setting (Agrawal, Maitra and Yamada, Crypto 2019). \smallskip In more detail, our results are: \begin{enumerate} \item We construct the first ABE for ${\sf NL}$ from the LWE, evasive LWE (Wee, Eurocrypt 2022 and Tsabary, Crypto 2022) and Tensor LWE (Wee, Eurocrypt 2022) assumptions. This yields the first (conjectured) post-quantum ABE for ${\sf NL}$. \item Relying on LWE, { evasive } LWE and a new assumption called {\it circular tensor} LWE, we construct ABE for all Turing machines. At a high level, the circular tensor LWE assumption incorporates circularity into the tensor LWE (Wee, Eurocrypt 2022) assumption. \end{enumerate} Towards our ABE for Turing machines, we obtain the first CP-ABE for circuits of unbounded depth and size from the same assumptions -- this may be of independent interest.

A {\it broadcast, trace and revoke} system generalizes broadcast encryption as well as traitor tracing. In such a scheme, an encryptor can specify a list $L \subseteq N$ of revoked users so that (i) users in $L$ can no longer decrypt ciphertexts, (ii) ciphertext size is independent of $L$, (iii) a pirate decryption box supports tracing of compromised users. The ``holy grail'' of this line of work is a construction which resists unbounded collusions, achieves all parameters (including public and secret key) sizes independent of $|L|$ and $|N|$, and is based on polynomial hardness assumptions. In this work we make the following contributions: 1. {\it Public Trace Setting:} We provide a construction which (i) achieves optimal parameters, (ii) supports embedding identities (from an exponential space) in user secret keys, (iii) relies on polynomial hardness assumptions, namely compact functional encryption (${\sf FE}$) and a key-policy attribute based encryption (${\sf ABE}$) with special efficiency properties, and (iv) enjoys adaptive security with respect to the revocation list. The previous best known construction by Nishimaki, Wichs and Zhandry (Eurocrypt 2016) which achieved optimal parameters and embedded identities, relied on indistinguishability obfuscation, which is considered an inherently subexponential assumption and achieved only selective security with respect to the revocation list. 2. {\it Secret Trace Setting:} We provide the first construction with optimal ciphertext, public and secret key sizes and embedded identities from any assumption outside Obfustopia. In detail, our construction relies on Lockable Obfuscation which can be constructed using ${\sf LWE}$ (Goyal, Koppula, Waters and Wichs, Zirdelis, Focs 2017) and two ${\sf ABE}$ schemes: (i) the key-policy scheme with special efficiency properties by Boneh et al. (Eurocrypt 2014) and (ii) a ciphertext-policy ${\sf ABE}$ for ${\sf P}$ which was recently constructed by Wee (Eurocrypt 2022) using a new assumption called {\it evasive and tensor} ${\sf LWE}$. This assumption, introduced to build an ${\sf ABE}$, is believed to be much weaker than lattice based assumptions underlying ${\sf FE}$ or ${\sf iO}$ -- in particular it is required even for lattice based broadcast, without trace. Moreover, by relying on subexponential security of ${\sf LWE}$, both our constructions can also support a {\it super-polynomial} sized revocation list, so long as it allows efficient representation and membership testing. Ours is the first work to achieve this, to the best of our knowledge.