## Homomorphic Public-Key Cryptosystems and Encrypting Boolean Circuits (2003)

Citations: | 16 - 5 self |

### BibTeX

@TECHREPORT{Grigoriev03homomorphicpublic-key,

author = {Dima Grigoriev and Ilia Ponomarenko},

title = {Homomorphic Public-Key Cryptosystems and Encrypting Boolean Circuits},

institution = {},

year = {2003}

}

### OpenURL

### Abstract

In this paper homomorphic cryptosystems are designed for the first time over any finite group. Applying Barrington's construction we produce for any boolean circuit of the logarithmic depth its encrypted simulation of a polynomial size over an appropriate finitely generated group.

### Citations

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Citation Context ...ural problems concerning secret computations is to design a homomorphic public-key cryptosystem over a finite group. The known examples of such systems include the quadratic residue cryptosystem (see =-=[12, 11]-=-) over the group of order 2 and the cryptosystems (see [22, 24, 25]) over some cyclic and dihedral groups. However, in these and some other cryptosystems the involved groups are solvable and so can no... |

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Citation Context ...Using such a cryptosystem one can efficiently implement a secret computation given by any circuit over the structure H. Some other applications of homomorphic public-key cryptosystems can be found in =-=[3, 8, 9, 27]-=-. We mention also that the group theory is a source of constructions (apart from homomorphic cryptosystems) in the cryptography, see e.g. [13, 16, 20, 21, 23]. It is well known that any boolean circui... |

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Citation Context ...Using such a cryptosystem one can efficiently implement a secret computation given by any circuit over the structure H. Some other applications of homomorphic public-key cryptosystems can be found in =-=[3, 8, 9, 27]-=-. We mention also that the group theory is a source of constructions (apart from homomorphic cryptosystems) in the cryptography, see e.g. [13, 16, 20, 21, 23]. It is well known that any boolean circui... |

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Citation Context ...Definitions and results. An important problem of modern cryptography concerns secret public-key computations in algebraic structures. There is a lot of public-key cryptosystems using groups (see e.g. =-=[2, 10, 11, 12, 14, 15, 16, 21, 22]-=- and also Subsection 1.3) but only a few of them have a homomorphic property in the sense of the following definition (cf. [11]). Definition 1.1 Let H be a finite nonidentity group, G a finitely gener... |

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Citation Context ... designed for the first time over finite commutative rings. 1 Introduction 1.1. The problem of constructing reliable cryptosystems for secret computations had been extensively studied last years (see =-=[3, 5, 10, 14, 26]-=-). Generally, it consists in encryption of a circuit over an algebraic structure H (e.g. group, ring, etc.). One of possible approaches to it is to find a publically known algebraic structure G and a ... |

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Citation Context ... designed for the first time over finite commutative rings. 1 Introduction 1.1. The problem of constructing reliable cryptosystems for secret computations had been extensively studied last years (see =-=[3, 5, 10, 14, 26]-=-). Generally, it consists in encryption of a circuit over an algebraic structure H (e.g. group, ring, etc.). One of possible approaches to it is to find a publically known algebraic structure G and a ... |

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Citation Context ...ogarithm problem [15]; the complexity of this system was studied in [4]. Some generalizations of this system to non-abelian groups (in particular, the matrix groups over some rings) were suggested in =-=[18]-=- where secrecy was based on an analog of the discrete logarithm problems in groups of inner automorphisms. Certain variations of the Diffie-Hellman systems over the braid groups were described in [12]... |

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Citation Context ...ow the output of B one has to be able to calculate f(g) 2 H, which is supposedly to be difficult due to Theorem 1.3. We mention that a different approach to encrypt boolean circuits was undertaken in =-=[24]-=-. 1.2. Discussion on complexity and security. One can see that the encryption procedure can be performed by means of public keys efficiently. However, the decryption procedure is a secret one in the f... |

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Citation Context ...ore one can easily verify the condition (H2) and on the other hand this allows one to provide evidence for the difficulty of a decryption. In this connection we mention a public-key cryptosystem from =-=[6]-=- in which f was the natural epimorphism from a free group G onto the group H (infinite, non-abelian in general) given by generators and relations. In this case for any element of H one can produce its... |

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Citation Context ... in this direction. In particular, we mention the cryptosystem from [7] based on a homomorphism from the direct sum of rings isomorphic Z. A finite version of this system [8] was 3recently broken in =-=[1]-=-. As the second main result of this paper we present a homomorphic public-key cryptosystem over a finite commutative ring (for details see Section 3). Before formulating it we recall that any finite c... |

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Citation Context ...Using such a cryptosystem one can efficiently implement a secret computation given by any circuit over the structure H. Some other applications of homomorphic public-key cryptosystems can be found in =-=[3, 8, 9, 27]-=-. We mention also that the group theory is a source of constructions (apart from homomorphic cryptosystems) in the cryptography, see e.g. [13, 16, 20, 21, 23]. It is well known that any boolean circui... |

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Citation Context ... designed for the first time over finite commutative rings. 1 Introduction 1.1. The problem of constructing reliable cryptosystems for secret computations had been extensively studied last years (see =-=[3, 5, 10, 14, 26]-=-). Generally, it consists in encryption of a circuit over an algebraic structure H (e.g. group, ring, etc.). One of possible approaches to it is to find a publically known algebraic structure G and a ... |

1 |
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Citation Context ...morphic public-key cryptosystems can be found in [3, 8, 9, 27]. We mention also that the group theory is a source of constructions (apart from homomorphic cryptosystems) in the cryptography, see e.g. =-=[13, 16, 20, 21, 23]-=-. It is well known that any boolean circuit of logarithmic depth can be efficiently simulated by a circuit over an arbitrary finite nonsolvable group, see [2] (another approach to encrypting boolean c... |