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Efficient Implementation of AND, OR and NOT Operators for ABCs

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Trusted Systems (INTRUST 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9565))

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Abstract

In the last few years several practitioners have proposed different strategies for implementing Attribute-based credentials (ABCs) on smart cards. ABCs allow citizens to prove certain properties about themselves without necessarily revealing their full identity. The Idemix ABC is the most versatile ABC system proposed in the literature, supporting peudonyms, equality proofs of representation, verifiable encryption of attributes and proving properties of attributes via AND, NOT and OR operators. Recently, Vullers et al. and De La Piedra et al. addressed the implementation of the selective disclosure operations, pseudonyms and multi-credential proofs such as equality proofs of representation. In this manuscript, we present implementation strategies for proving properties of user attributes via these operators and show how to combine them via external and internal commitment reordering.

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Notes

  1. 1.

    https://www.irmacard.org.

  2. 2.

    Our performance figures have been extracted relying on a MULTOS ML3-R3-80K smart card using the SCM Microsystems SCL011 reader in a Intel Core i5-3230M CPU clocked at 2.60 GHz running Debian Linux 3.13.6-1, python 2.7.6, python-pyscard 1.6.12.1-4 and CHARM 0.43 [2].

  3. 3.

    In the first stage, the prover sends to the verifier a commitment message t or t_value. In the second move, the verifier sends to the prover a random challenge message c. Finally, the last message sent by the prover includes a response value or s_value.

  4. 4.

    For instance, an empty proof of possession over a set of attributes \((m_0, ..., m_5)\) is represented using the Camenisch-Staedler notation [11] as: NIZK \(:\{(\varepsilon ', \nu ', \alpha _0,...,\alpha _5): Z \equiv \pm R_0^{\alpha _0}R_1^{\alpha _1}R_2^{\alpha _2} R_3^{\alpha _3}R_4^{\alpha _4}R_5^{\alpha _5}A^{\varepsilon '}S^{\nu '} ~\mathrm {mod}~n \}\) being the Greek letters \((\varepsilon ', \nu ')\) and \((\alpha _0,...,\alpha _5)\) the values of the signature and the set of attributes proved in zero knowledge and not revealed.

  5. 5.

    As described in [18], the attributes are represented as \(l_m = 256\) bits. The rest of parameters are set as \(l'_{e} = 120\) (size of the interval where the e values are selected), (security parameter of the statistical ZKP), \(l_H = 256\) (domain of the hash function in the Fiat-Shamir heuristic), \(l_e = 504\) (size of e), \(l_n = 1,024\) (size of the RSA modulus) and \(l_v = 1,604\) bits (size of v).

  6. 6.

    http://www.infineon.com/dgdl/SPO_SLE+78CX1280P_2012-07.pdf?folderId=db3a304325afd6e00126508d47f72f66&fileId=db3a30433fcce646013fe3d672214ab8 (Accessed 27 February 2015).

  7. 7.

    Thus, for one possibility per attribute, we prove the non-existence of one attribute in \(m_i\). In this case, \(m_i = 3\) and \(m_t = 5\cdot 7\cdot 11\cdot 13\) (case 1). We consider 10 possibilities per attribute (50 primes). We prove the non-existence of one attribute in \(m_i\). For \(m_i = 3\), \(m_t = 179\cdot 181\cdot 191\cdot 193\) (case 2). We consider 1,000 possibilities per attribute (i.e. 5,000 primes) and we prove the non-existence of two attributes in \(m_t\) for \(m_i = 1,999\cdot 2,161\) and \(m_t = 3,323\cdot 3,253\cdot 2,897\cdot 2,999\) (case 3). Finally, we consider 10,000 possibilities per attribute (50,000 primes) and we proof the non-existence of two primes \(m_i = 91,387\cdot 91,393\) in \(m_t = 102,461\cdot 102,481\cdot 102,497\cdot 102,499\) (case 4).

  8. 8.

    We use the following notation in Tables 4, 5 and 6: PRE means precomputation, EUC 1-3 is related to the cases presented in Table 3, RA means Reveal all the Attributes with the exception of the master secret and HA to hide every attribute in the credential.

  9. 9.

    In this manuscript we only address the first version of this NIZK described in [7] and leave the second one beyond the scope of this work due the computation limitations of our target device.

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Authors and Affiliations

  1. ICIS DS, Radboud University Nijmegen, Nijmegen, The Netherlands

    Antonio de la Piedra

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  1. Antonio de la Piedra
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Correspondence to Antonio de la Piedra .

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Editors and Affiliations

  1. Computer Science, Columbia University, New York, New York, USA

    Moti Yung

  2. Key Laboratory of Trusted Computing, Beijing University of Technology, Beijing, China

    Jianbiao Zhang

  3. Key Laboratory of Trusted Computing, Beijing University of Technology, Beijing, China

    Zhen Yang

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de la Piedra, A. (2016). Efficient Implementation of AND, OR and NOT Operators for ABCs. In: Yung, M., Zhang, J., Yang, Z. (eds) Trusted Systems. INTRUST 2015. Lecture Notes in Computer Science(), vol 9565. Springer, Cham. https://doi.org/10.1007/978-3-319-31550-8_12

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