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Circular and KDM Security for Identity-Based Encryption

Jacob Alperin-Sheriff Chris Peikert

May 1, 2012

Abstract

We initiate the study of security for key-dependent messages (KDM), sometimes also known ascircular or clique security, in the setting of identity-based encryption (IBE). Circular/KDM securityrequires that ciphertexts preserve secrecy even when they encrypt messages that may depend on the secretkeys, and arises in natural usage scenarios for IBE.

We construct an IBE system that is circular secure for affine functions of users secret keys, based onthe learning with errors (LWE) problem (and hence on worst-case lattice problems). The scheme is securein the standard model, under a natural extension of a selective-identity attack. Our three main technicalcontributions are (1) showing the circular/KDM-security of a dual-style LWE public-key cryptosystem,(2) proving the hardness of a version of the extended LWE problem due to ONeill, Peikert and Waters(CRYPTO11), and (3) building an IBE scheme around the dual-style system using a novel lattice-basedall-but-d trapdoor function.

1 Introduction

Traditional notions of secure encryption, starting with semantic (or IND-CPA) security [GM82], assume thatthe plaintext messages do not depend on the secret decryption key (except perhaps indirectly, via the publickey or other ciphertexts). In many settings, this may fail to be the case. One obvious scenario is, of course,careless or improper key management: for example, some disk encryption systems end up encrypting thesymmetric secret key itself (or a derivative) and storing it on disk. However, there are also situations in whichkey-dependent messages are used as an integral part of an cryptosystem. For example, in their anonymouscredential system, Camenisch and Lysyanskaya [CL01] use a cycle of key-dependent messages to discourageusers from delegating their secret keys. More recently, Gentrys bootstrapping technique for obtaininga fully homomorphic cryptosystem [Gen09] encrypts a secret key under the corresponding public key inorder to support unbounded homomorphism; the cryptosystem therefore needs to be circular secure. Moregenerally, a system that potentially reveals encryptions of any partys secret key under any users public keyneeds to be clique or key-dependent message (KDM) secure. This notion allows for proving formalsymbolic soundness of cryptosystems having complexity-based security proofs [ABHS05].

Since Boneh et al.s breakthrough work [BHHO08] giving a KDM-secure encryption scheme, in thestandard model, from the Decision Diffie-Hellman assumption, a large number of results (mostly positive)School of Computer Science, College of Computing, Georgia Institute of Technology. Email: jmas6@cc.gatech.eduSchool of Computer Science, Georgia Institute of Technology. Email: cpeikert@cc.gatech.edu. This material is based

upon work supported by the National Science Foundation under Grant CNS-0716786 and CAREER Award CCF-1054495, and bythe Alfred P. Sloan Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those ofthe authors and do not necessarily reflect the views of the National Science Foundation or the Sloan Foundation.

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have been obtained regarding circular- and KDM-secure encryption [HH09, ACPS09, BHHI10, BG10,App11, MTY11, BGK11, BV11]. However, all these works have dealt only with the symmetric or public-keysettings; in particular, the question of circular/KDM security for identity-based cryptography has not yet beenconsidered. Recall that in identity-based encryption [Sha84], any string can serve as a public key, and thecorresponding secret keys are generated and administered by a trusted Private Key Generator (PKG).

Circular security for IBE. In this work we define and construct a circular/KDM-secure identity-basedencryption (IBE) scheme. KDM security is well-motivated by some natural usage scenarios for IBE, as wenow explain.

Recall that identity-based encryption gives a natural and lightweight solution to revocation, via expiringkeys. The lifetime of the cryptosystem is divided into time periods, or epochs. An identity string consistsof a users true identity (e.g., name) concatenated with an epoch; when encrypting, one uses the identity forthe current epoch. To support revocation, the PKG gives out a users secret key only for the current epoch,and only if the user is still authorized to be part of the system. Therefore, a users privileges can be revokedby simply refusing to give out his secret key in future epochs; in particular, this revocation is transparent tothe encrypter.

The above framework makes it necessary for users to periodically get new secret keys from the PKG,confidentially. The most lightweight method, which eliminates the need for the user to prove his identityevery time, is simply for the PKG to encrypt the new secret key under the users identity for the previousepoch. This can be proved secure, assuming the underlying IBE is CPA-secure, as long as there are no cyclesof encrypted keys. However, if a user deletes or loses an old secret key and wants to decrypt a ciphertext fromthe corresponding epoch, it would be natural for the authority to provide the old secret key encrypted underthe users identity for the current epoch. But because the current secret key has also been encrypted (perhapsvia a chain of encryptions) under the old identity, this may be unsafe unless the IBE is KDM-secure.

1.1 Our Contributions

As already mentioned, in this work we define a form of circular/KDM security for identity-based encryption,and give a standard-model construction based on the learning with errors (LWE) problem, hence on worst-caselattice problems via the reductions of [Reg05, Pei09].

As in prior positive results on circular security [BHHO08, ACPS09, BG10], our definition allows theadversary to obtain encrypted key cliques for affine functions of the users secret keys. More precisely, forany tuple of identities (id1, . . . , idd), the attacker may adaptively query encryptions of f(skid1 , . . . , skidd)under any of the identities idj , where f is any affine function over the message space, and each skidi is asecret key for identity idi. (This obviously specializes to encryptions of a single secret key.) Our attackmodel is in the style of a selective identity attack, wherein the adversary must declare the target identitiesid1, . . . , idd (but not the functions f ) before seeing the public parameters of the scheme. While this is not thestrongest security notion we might hope for, it appears to at least capture the main security requirements ofthe scenarios described above, because encrypted key cycles are only ever published for the same real-worldidentity at different time epochs. Therefore, just as in a standard selective-identity attack for IBE, theadversary is actually limited to attacking just a single real-world identity, on a set of d epochs (which could,for example, include all valid epochs). We also point out that by a routine hybrid argument, security also holdswhen attacking a disjoint collection of identity cliques (that are named before seeing the public parameters).

Our IBE construction is built from two components, both of which involve some novel techniques. First,we give an LWE-based public-key cryptosystem that is clique secure (even for an unbounded number of users)

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for affine functions, and is suitable for embedding into an IBE like the one of [GPV08]. Second, we constructa lattice-based all-but-d trapdoor function that serves as the main foundation of the IBE. We elaborate onthese two contributions next.

Clique-secure public-key cryptosystem. We first recall that Applebaum et al. [ACPS09] showed thata variant of Regevs so-called primal LWE cryptosystem [Reg05] is clique secure for affine functions.Unfortunately, this primal-type system does not seem suitable as the foundation for identity-based encryptionusing the tools of [GPV08]. The first reason is that the proof of clique security from [ACPS09] needs theusers public keys to be completely independent, rather than incorporating a shared random string (like thepublic parameters in an IBE system). The second reason is a bit more technical, and is already describedin [GPV08]: in primal-style systems, the user-specific public keys are exponentially sparse pseudorandomvalues (with unique secret keys), and it is difficult to design an appropriate mapping from identities to validpublic keys that actually admit usable secret keys.

Therefore, we first need to obtain clique security for a so-called dual-type cryptosystem (using theterminology from [GPV08]), in which every syntactically valid public key has a functional secret key (actually,many such secret keys) that can be extracted by the PKG. It turns out that achieving this goal is quite abit more technically challenging than it was for the primal-style schemes. This is primarily because theKDM-secure scheme from [ACPS09] (like the earlier one from [BHHO08]) has the nice property that giventhe public key alone, one can efficiently generate statistically well-distributed encryptions of the secret key(without knowing the corresponding encryption randomness). This immediately implies circular security forself-loops, and clique security follows from a couple of other related techniques.

Unfortunately, this nice statistical property on ciphertexts does not seem attainable for dual-style LWEencryption, because now valid ciphertexts are exponentially sparse and hard to generate without knowing theunderlying encryption randomness. In addition, because the adversary may obtain an unbounded number ofkey-dependent ciphertexts, we also cannot rely on any statistical entropy of the secret key conditioned on thepublic key, as is common in the security proofs of