New Explicit Conditions of Elliptic Curve Traces for FR-Reduction

By Atsuko Miyaji, Shunzo Takano. 2001

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Citations in the corpus

Geppetto: Versatile Verifiable Computation (2015, background): “Remarkably, Karabina and Teske [35] show that it is possible to generate two MNT curves [45] E and ˜ E that are pairing friendly and, more importantly, ˜ E can be embedded in E , and E can be embedded in ˜ E .”
Short Group Signatures (2004, methodology): “However, we allow for the more general case where G 1 = G 2 so that our constructions can make use of certain families of nonsupersingular elliptic curves defined by Miyaji et al. [28].”
EASiER: encryption-based access control in social networks with efficient revocation (2011, methodology): “The implementation uses MNT curves [11] with a 159bit base field.”
Short Signatures from the Weil Pairing (2001, background): “The first, based on [41], is a family of non-supersingular curves over a prime finite field.”
Proxy Provable Data Possession in Public Clouds (2013, methodology): “Such a bilinear map ^ e can be constructed by the modified Weil [24] or Tate pairings [25] on elliptic curves.”
Public Key Encryption with Conjunctive Keyword Search and Its Extension to a Multi-user System (2007, methodology): “We can use the fact that G1 is of size approximately 2170, elements in G1 are 171-bit strings, and the discrete log in G1 is as hard as the discrete log in Zq where q is 1020 bits [10,15,28].”
A ciphertext-policy attribute-based encryption scheme with constant ciphertext length (2009, background): “Although a symmetric bilinear map is required in these proofs, our schemes can be proven with an asymmetric bilinear map such as the Weil or Tate pairing e : G1×G2 → GT over MNT curves [24], where G1 and G2 are distinct groups.”
Secure attribute-based systems (2006, methodology): “For instance, systems using our API can choose between the two most studied elliptic curve groups providing bilinear maps: supersingular elliptic curves (SS), which enable fast cryptographic pairing operations [30], and MNT elliptic curves, which are used to obtain small ciphertext sizes [31].”
Compact E-Cash (2005, methodology): “It has been conjectured that this property may hold when the bilinear groups are instantiated using either the Weil or Tate pairing over MNT curves [47][7, Sec.”
Pairing-Friendly Elliptic Curves of Prime Order (2005, methodology): “The pioneering work of Miyaji, Nakabayashi, and Takano [15] describes how to construct elliptic curves of prime order and embedding degree k ∈ {3, 4, 6}.”
jPBC: Java pairing based cryptography (2011, methodology): “3) Type D: By using the CM method, see [9] for a survey on this method, it is possible to construct curves with embedding degree 6, see [10], [11].”
ECPP: Efficient Conditional Privacy Preservation Protocol for Secure Vehicular Communications (2008, methodology): “Table II gives the measured processing time (in milliseconds) for an MNT curve [19] of embedding degree k = 6 and 160-bit q.”
Efficient and Provably-Secure Identity-Based Signatures and Signcryption from Bilinear Maps (2005, background): “Such bilinear map groups are known to be instantiable with ordinary elliptic curves such as those suggested in [29] or [4].”
On Prime-Order Elliptic Curves with Embedding Degrees k = 3, 4, and 6 (2008, methodology): “From (1) Miyaji, Nakabayashi, and Takano [15] derived Pell-type equations, which we subsequently call MNT equations (see Section 2).”
A One Round Protocol for Tripartite Diffie–Hellman (2000, background): “These constructions are described in [22] and [35].”
Hierarchical Identity Based Encryption with Constant Size Ciphertext (2005, background): “Such maps let us take advantage of certain curves called MNT curves [20].”