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A signature scheme from the finite field isomorphism problem. (English) Zbl 1450.94051
Let $$F_q$$ be the finite field with $$q$$ elements, and let $$f(x)\in F_q[x]$$ and $$F(y)\in F_q[y]$$ be irreducible monic polynomials of degree $$n$$. We set $$X=F_q[x]/f(x)$$ and $$Y = F_q[y]/f(y)$$, and consider an isomorphism $$\phi : X \rightarrow Y$$. Let $$1 \leq \beta \leq q/2$$ be a parameter, and denote by $$X[\beta]$$ the set of $$a(x)\in X$$ with $$L^{\infty}$$-norm bounded by $$\|a\| \leq \beta$$. We select $$a_1(x), \ldots, a_k(x)$$ uniformly and randomly from $$X[\beta]$$, and let $$A_i = \phi(a_i)$$ $$(i=1,\ldots,k)$$ be the corresponding images in $$Y$$. In [Y. Doröz et al., Lect. Notes Comput. Sci. 10769, 125–155 (2018; Zbl 1385.94032)] a new hard problem, the Finite Field Isomorphism Problem (FFI) is presented. More precisely, we have the two following versions of this problem:
The Computational FFI problem (CFFI): Given $$Y$$ and the list of polynomials $$A_1(y), \ldots, A_k(y)$$, recover $$f(x)$$ and/or one or more of $$a_1(x), \ldots, a_k(x)$$.
The Decisional FFI problem (DFFI): Let $$\epsilon > 0$$. Let $$b_1(x)$$ be randomly chosen in $$X[\beta]$$, let $$B_1(y) = \phi(b_1)$$, and let $$B_2(y)$$ be randomly chosen in $$Y$$. Given the data $$Y, A_1(y), \ldots, A_k(y)$$ and the pair $$\{B_1(y), B_2(y)\}$$ in a random order, identify, with probability greater than $$1/2+ \epsilon$$, which element of the pair was constructed using $$\phi$$.
In the aforementioned paper, a new fully homomorphic encryption scheme from the DFFI problem is given. In the paper under review, a signature scheme from the CFFI problem is constructed. For any polynomial $$h(x) \in X$$, the associated NTRU lattice is defined to be the $$2n$$-dimensional lattice $$L_h$$ formed by the pairs $$(u(x), v(x))\in Z[x]^2$$ with $$\deg u \leq n-1$$, $$\deg v \leq n-1$$ and $$v(x) \equiv u(x)h(x)\pmod{(q,f(x))}$$ and similarly for $$L_{\phi(h)}$$. The proposed signature scheme is build via the following framework:
1.
Generate a signature $$s$$, which is a short vector within or close to a lattice $$L_h$$ related to the hidden field $$X$$.
2.
Publish its image $$S \in Y$$, and prove the validity of the signature by showing a relationship between $$S$$ and a lattice $$L_{\phi(h)}$$ related to the public field $$Y$$.

The assumption that the map $$\phi : X \rightarrow Y$$ behaves randomly, implies that there is a negligible probability that the public lattice $$L_{\phi(h)}$$ will have any exceptionally short vectors. Thus, trapdoors can be build using short vectors in $$L_h$$ without the necessity of concealing the trapdoor from the attacker. It follows that one can use very efficient methods to generate $$s$$. The key idea is that the attacker is not allowed to see the lattice $$X$$, which contains one or more vectors that are considerably shorter than predicted by the Gaussian heuristic.
##### MSC:
 94A62 Authentication, digital signatures and secret sharing 94A60 Cryptography
##### Software:
BKZ; BLISS; Magma; NTRU; NTRUSign
Full Text:
##### References:
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