Cryptanalyses of Branching Program Obfuscations over GGH13 Multilinear Map from the NTRU Problem

2018 ◽  
pp. 184-210 ◽  
Author(s):  
Jung Hee Cheon ◽  
Minki Hhan ◽  
Jiseung Kim ◽  
Changmin Lee
Keyword(s):  
Multilinear Map

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

An alternative proof on higher order derivatives of a multilinear map

2018 ◽  
Vol 68 (7) ◽  
pp. 1457-1464
Author(s):  
Sónia Carvalho
Keyword(s):  
Higher Order ◽  
Alternative Proof ◽  
Multilinear Map ◽  
Derivatives Of

scholarly journals An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero

2016 ◽  
Vol 19 (A) ◽  
pp. 255-266 ◽  
Author(s):  
Jung Hee Cheon ◽  
Jinhyuck Jeong ◽  
Changmin Lee
Keyword(s):  
Polynomial Time ◽  
Euclidean Norm ◽  
Low Level ◽  
Ideal Lattices ◽  
Security Parameter ◽  
Main Technique ◽  
Multilinear Maps ◽  
Multilinear Map

Let$\mathbf{f}$and$\mathbf{g}$be polynomials of a bounded Euclidean norm in the ring$\mathbb{Z}[X]/\langle X^{n}+1\rangle$. Given the polynomial$[\mathbf{f}/\mathbf{g}]_{q}\in \mathbb{Z}_{q}[X]/\langle X^{n}+1\rangle$, the NTRU problem is to find$\mathbf{a},\mathbf{b}\in \mathbb{Z}[X]/\langle X^{n}+1\rangle$with a small Euclidean norm such that$[\mathbf{a}/\mathbf{b}]_{q}=[\mathbf{f}/\mathbf{g}]_{q}$. We propose an algorithm to solve the NTRU problem, which runs in$2^{O(\log ^{2}\unicode[STIX]{x1D706})}$time when$\Vert \mathbf{g}\Vert ,\Vert \mathbf{f}\Vert$, and$\Vert \mathbf{g}^{-1}\Vert$are within some range. The main technique of our algorithm is the reduction of a problem on a field to one on a subfield. The GGH scheme, the first candidate of an (approximate) multilinear map, was recently found to be insecure by the Hu–Jia attack using low-level encodings of zero, but no polynomial-time attack was known without them. In the GGH scheme without low-level encodings of zero, our algorithm can be directly applied to attack this scheme if we have some top-level encodings of zero and a known pair of plaintext and ciphertext. Using our algorithm, we can construct a level-$0$encoding of zero and utilize it to attack a security ground of this scheme in the quasi-polynomial time of its security parameter using the parameters suggested by Garg, Gentry and Halevi [‘Candidate multilinear maps from ideal lattices’,Advances in cryptology — EUROCRYPT 2013(Springer, 2013) 1–17].

An Improved Multilinear Map and its Applications

2015 ◽  
Vol 10 (3) ◽  
pp. 64-81 ◽  
Author(s):  
Chunsheng Gu
Keyword(s):  
Random Matrix ◽  
Principal Ideal ◽  
Key Exchange ◽  
Ideal Lattice ◽  
Witness Encryption ◽  
Lattice Problem ◽  
Multilinear Maps ◽  
Multilinear Map ◽  
Cover Problem ◽  
Exact Cover

Cryptographic multilinear maps have extensive applications. However, current constructions of multilinear maps suffer from the zeroizing attacks. For a candidate construction of multilinear maps described by Garg, Gentry, and Halevi (GGH13), Hu & Jia recently presented an efficient attack, which broke the GGH13-based applications of multipartite key exchange (MPKE) and witness encryption (WE) based on the hardness of 3-exact cover problem. By introducing random matrix, the author presents an improvement of the GGH13 map, which supports the applications for public tools of encoding in the GGH13 map, such as MPKE and WE. The security of the construction depends upon new hardness assumption. Moreover, the author's improvement destroys the structure of the ring element in the principal ideal lattice problem, and avoids potential attacks using algorithm of solving short principal ideal lattice generator.

The Mediating Morphism of the Multilinear Optimal Map

2014 ◽  
Vol 4 (1) ◽  
pp. 82-87
Author(s):  
Seak-Weng Vong ◽  
Xiao-Qing Jin ◽  
Jin-Hua Wang
Keyword(s):  
Tensor Product ◽  
Short Note ◽  
Linear Transform ◽  
Multilinear Map ◽  
Definition Of ◽  
Product Of Matrices

AbstractIn this short note, we study a relation between the tensor product of matrices and a multilinear map defined by the optimal operator. In this particular case, the linear transform (mediating morphism) hidden in the abstract definition of the general tensor product can be determined explicitly.

scholarly journals Calculations on Lie Algebra of the Group of Affine Symplectomorphisms

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Zuhier Altawallbeh
Keyword(s):  
Lie Algebra ◽  
Homological Algebra ◽  
Hochschild Homology ◽  
Exterior Algebra ◽  
String Topology ◽  
Potential Interest ◽  
Multilinear Map ◽  
Leibniz Homology ◽  
Algebra Homology

We find the image of the affine symplectic Lie algebra gn from the Leibniz homology HL⁎(gn) to the Lie algebra homology H⁎Lie(gn). The result shows that the image is the exterior algebra ∧⁎(wn) generated by the forms wn=∑i=1n(∂/∂xi∧∂/∂yi). Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology HH⁎-1(U(gn)) is isomorphic to H⁎-1Lie(gn,U(gn)ad). Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.

Sign in / Sign up

Export Citation Format

Share Document