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Universally Composable Identity-Based Encryption

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Universally Composable Identity-Based Encryption SCIS 2006 The 2006 Symposium on Cryptography and Information Security All rights are reserved and copyright of this manuscript belongs to the authors. Hiroshima, Japan, Jan. 17-20, 2006 This manuscript has been published without reviewing and editing as received The Institute of Electronics, from the authors: posting the manuscript to SCIS 2006 does not prevent future Information and Communication Engineers submissions to any journals or conferences with proceedings. Universally Composable Identity-Based Encryption Ryo Nishimaki ¤ Yoshifumi Manabe ¤ y Tatsuaki Okamoto ¤ y Abstract| The identity-based encryption (IBE) is one of the most important primitives in cryp- tography, and various security notions of IBE (e.g., IND-ID-CCA2, NM-ID-CCA2, IND-sID-CPA etc.) have been introduced and the relations among them have been clari¯ed recently. This paper, for the ¯rst time, investigate the security of IBE in the universally composable (UC) framework. This paper ¯rst de¯nes the UC-security of IBE, i.e., we de¯ne the ideal functionality of IBE, FIBE. We then show that UC-secure IBE is equivalent to conventionally-secure (IND-ID-CCA2-secure) IBE. This paper also introduces the UC-security of weaker security notions of IBE, which correspond to IND-ID-CPA IBE and IND-sID-CCA2. We ¯nally prove that Boneh-Franklin's suggestion on the construction of a secure signatures from an IND-ID-CPA IBE scheme is true in the UC framework. Keywords: identity-based encryption, IND-ID-CCA2, universal composition, digital signatures 1 Introduction entionally-secure primitives/protocols has been one of the signi¯cant topics in cryptography [4, 5, 8, 9, 11]. 1.1 Background Since UC represents stronger security requirements, a The concept of identity-based encryption (IBE) was lot of conventionally-secure protocols fail to meet UC introduced by Shamir [12], and is a variant of public- security requirements. For example, we cannot design key encryption (PKE), where the identity of a user is secure two party protocols in the UC framework with employed in place of the user's public-key. no setup assumption, while there are conventionally- Boneh and Franklin [2] de¯ned the security, IND- secure two party protocols (e.g., commitment and zero- ID-CCA2 (indistinguishable against adaptively chosen- knowledge proofs) with no setup assumption. ciphertext attacks under chosen identity attacks), as However, we know that the conventional security the desirable security of IBE schemes. Canetti, Halevi, notions are equivalent to UC security notions for a and Katz [6, 7] de¯ned a weaker notion of security in few cryptographic primitives. For example, UC-secure which the adversary commits ahead of time to the chal- PKE is equivalent to conventionally-secure (IND-CCA2- lenge identity it will attack. We refer to this notion as secure) PKE [3] and UC-secure signatures are equiva- selective identity (sID) adaptively chosen-ciphertext se- lent to conventionally-secure (existentially unforgeable cure IBE (IND-sID-CCA2). In addition, they also de- against chosen message attacks: EUF-CMA-secure) sig- ¯ne a weaker security notion of IBE, selective-identity natures [4]. chosen-plaintext (CPA) secure IBE (IND-sID-CPA). At- IBE is a more complex cryptographic primitive than trapadung et. al. [1], and Galindo and Hasuo [10] PKE and signatures, so it is not clear whether conventi- introduced the non-malleability (NM) in the security onally-secure (i.e., IND-ID-CCA2-secure) IBE is equiv- notion of IBE. Thus, the security de¯nitions consid- alent to UC-secure IBE or not. Since IBE is one of the ered up to now in the literature are: G-A1-A2, where most signi¯cant primitives like PKE and signatures in G 2 fIND, NMg, A1 2 fID, sIDg, ID denotes full- cryptography, it is important to clarify the relationship identity attacks, and A2 2 fCPA, CCA1, CCA2g. between the UC security and conventional security no- Attrapadung et. al. [1], and Galindo and Hasuo tions of IBE. The UC security of IBE, however, has not [10] have clari¯ed the relationship among these no- been investigated. tions, and shown that IND-ID-CCA2 is equivalent to That is, we have the following problems: the strongest security notion, NM-ID-CCA2, among them. 1. What is the security de¯nition of IBE in the UC Since Canetti introduced universal composability (UC) framework (i.e., how to de¯ne an ideal function- as a new framework for analyzing the security of cryp- ality of IBE)? tographic primitives/protocols [3], investing the rela- 2. Is UC-secure IBE equivalent to IND-ID-CCA2- tion between UC-secure primitives/protocols and conv- secure IBE? ¤ Department of Social Informatics, Graduate School of Infor- Some weaker security notions of IBE than IND-ID- matics, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto- CCA2 are also useful to construct a secure (IND-CCA2) shi, Japan, [email protected] y NTT Laboratories, 1-1 Hikari-no-oka, Yokosuka- PKE scheme and a secure (EUF-CMA) signatures. For shi, Japan, [email protected], example, Canetti, Halevi and Katz [7] have shown how [email protected] to construct a secure PKE scheme from a selective-ID- We say that a function g : < ! < is negligible if for j j 1 secure (IND-sID-CPA-secure) IBE scheme. Boneh and any d > 0 we have g(k) < kd for su±ciently large k. Franklin [2] suggested a construction of a secure signa- tures from an IND-ID-CPA IBE scheme. 2.2 Identity-Based Encryption The UC security treatment of such weaker security Identity-Based Encryption scheme Identity-based notions of IBE may provide insight into a new relation- encryption scheme § is speci¯ed by four algorithms: S, ship between IBE and other primitives and also o®er a X , E, D: simpler and more clear proof of the relations than the Setup: S takes security parameter k and returns params conventional proofs. Therefore, it should be signi¯cant (system parameters) and mk (master-key). The to de¯ne the weaker security notions of IBE in the UC system parameters include a description of a ¯- framework. nite message space M, and a description of a ¯- That is, we have the following problem: nite ciphertext space C. 1. What are the UC security de¯nitions of the weaker Extract: X takes as input params, mk, and an ar- security notions of IBE? bitrary ID 2 f0; 1g¤, and returns private key d. 2. How to prove the constructibility of secure PKE/ Here ID is an arbitrary string that will be used signatures from the weaker security notions of as a public key, and d is the corresponding private IBE in the UC framework? decryption key. E 2 M 1.2 Our Results Encrypt: takes as input params, ID, and M . It returns a ciphertext C 2 C. This paper answers the above-mentioned problems: Decrypt: D takes as input params, C 2 C, and a 1. This paper de¯nes the UC-security of IBE, i.e., private key d. It returns M 2 M. we de¯ne the ideal functionality of IBE, FIBE. These algorithms must satisfy the standard consis- 2. We show that UC-secure IBE is equivalent to tency constraint, namely, conventionally-secure (IND-ID-CCA2-secure) IBE. 8M 2 M : D(params; C; d) = M where C = 3. We de¯ne the ideal functionalities of weaker se- E(params; ID; M), d = X (params; mk; ID) F ND F sID curity notions of IBE, IBE and IBE. We then F ND 2.3 De¯nitions of security notions for IBE schemes show that UC-secure IBE with IBE is equivalent to IND-ID-CPA IBE, and that UC-secure IBE Let A = (A1; A2) be an adversary; we say A is poly- F sID nomial time if both probabilistic algorithm A and A with IBE is equivalent to IND-sID-CCA2 IBE. 1 2 are polynomial time. At the ¯rst stage, given the sys- 4. We prove that Boneh-Franklin's suggestion [2] tem parameters, the adversary computes and outputs on the construction of a secure signatures from challenge template ¿. A1 can output some information an IND-ID-CPA IBE scheme is true in the UC s which will be transferred to A2. At the second stage, framework. That is, we present a protocol which the adversary is challenged with ciphertext y¤ gener- F F ND UC-realizes ideal functionality SIGW in the IBE- ated from ¿ by a probabilistic function, in a manner F hybrid model, where SIGW is an ideal function- depending on the goal. We say adversary A success- F ality with (normal) unforgeability, while SIG de- fully breaks the scheme if she achieves her goal. We ¯ned by Canetti [4] represents strong unforgeabil- consider a security goal, IND [1, 10], and three attack ity. models, ID-CPA, ID-CCA, ID-CCA2, listed in order of increasing strength. The di®erence among the models 2 Preliminaries is whether or not A1 or A2 is granted access to decryp- 2.1 Conventions tion oracles. We describe in Table 1 the ability with which the adversary can, in the di®erent attack mod- Notations We describe probabilistic algorithms and els, access the Extraction Oracle X (params; mk; ¢), the experiments with standard notations and conventions. Encryption Oracle E(params; ID; ¢) and the Decryp- For probabilistic algorithm A, A(x1; x2; :::; r) is the re- tion Oracle D(params; d; ¢) (We omit parameters in sult of running A that takes as inputs x1; x2; ::: and Table 1). When we say O = fEO ; XO ; DO g = Ã i i i i coins r.
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