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Encyclopedia of Cryptography and Security pp 51–52Cite as

Blum–Goldwasser Public Key Encryption System

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The Blum–Goldwasser public key encryption system combines the general construction of Goldwasser–Micali [5] with the concrete Blum– Blum–Shub pseudorandom bit generator [2] to obtain an efficient semantically secure public key encryption whose security is based on the difficulty of factoring Blum integers. The system makes use of modular arithmetic and works as follows:

  • Key Generation. Given a security parameter τ∈ℤ as input, generate two random τ-bit primes \(p,q\) where \(p=q=3 \bmod 4\). Set \(N=pq \in Z\). The public key is N and private key is \((p,q)\).

  • Encryption. To encrypt a message \(m = m_1 \ldots m_\ell \in \{0,1\}^\ell\):

    1. 1.

      Pick a random x in the group N * N and set x 1=x 2∈ℤ * N .

    2. 2.

      For \(i=1,\ldots,\ell\):

      1. (a)

        View \(x_i\) as an integer in \([0,N-1]\) and let \(b_i \in \{0,1\}\) be the least significant bit of \(x_i\).

      2. (b)

        Set c i =m i =m i ⊕b i ∈{0, 1}.

      3. (c)

        Set x i+1=x 2 i ∈ℤ * N .

    3. 3.

      Output (c 1, ..., c ℓ, x ℓ+1)∈{0, 1}ℓ×ℤ M as the ciphertext.

      ...

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References

  1. Bellare, Mihir and Phillip Rogaway (1996). “The exact security of digital signatures: How to sign with RSA and Rabin.” Advances in Cryptology—EUROCRYPT'96, Lecture Notes in Computer Science, vol. 1070, ed. U. Maurer. Springer-Verlag, Berlin, 399–416.

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  2. Blum, L., M. Blum, and M. Shub (1983). “Comparison of two pseudo-random number generators.” Advances in Cryptology—CRYPTO'83, ed. D. Chaum. Springer-Verlag, Berlin, 61–78.

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  3. Boneh, Dan (2001). “Simplified OAEP for the RSA and Rabin functions.” Advances in Cryptology—CRYPTO 2001, Lecture Notes in Computer Science, vol. 2139, ed. J. Kilian. Springer-Verlag, Berlin.

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  4. Fujisaki, E. and T. Okamoto (1999). “Secure integration of asymmetric and symmetric encryption schemes.” Advances in Cryptology—CRYPTO'99, Lecture Notes in Computer Science, vol. 1666, ed. J. Wiener. Springer-Verlag, Berlin, 537–554.

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  5. Goldwasser, S. and S. Micali (1984). “Probabilistic encryption.” Journal of Computer and System Science (JCSS), 28 (2), 270–299.

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Authors
  1. Dan Boneh
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Editor information

Editors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

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© 2005 International Federation for Information Processing

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Boneh, D. (2005). Blum–Goldwasser Public Key Encryption System. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_38

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