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Identification and signatures based on NP-hard problems of indefinite quadratic forms. (English) Zbl 1163.11027
Some problems related to quadratic forms are proved NP-hard under probabilistic reductions. Any bilinear form $$f(X)=\sum_{1\leq i\leq j\leq n}a_{ij}X_iX_j$$ with integer coefficients is represented by a symmetric matrix $$A$$ such that $$f(X)=X^TAX$$. Let us write in this review $$A=A_f$$ and $$f=f_A$$. two forms $$f,g$$ are equivalent if $$A_g=T^TA_fT$$ for some $$T\in\text{GL}(n)$$, $$T$$ with integer entries. In this case, it is written $$g=fT$$. A form $$f$$ represents an integer $$y$$ if there exists an integer vector $$x$$ such that $$f(x)=y$$ and the representation is primitive if the entries of $$x$$ have 1 as gcd.
The computational equivalence problem (CEP) consists in finding $$T$$ such that $$g=fT$$ given a pair $$(f,g)$$ of equivalent forms. The computational representation problem (CR) consists in finding a primitive representation $$x$$ of an integer $$y$$ (i.e. in solving in the integers the equation $$y=f(x)$$) given a form $$f$$ and the integer $$y$$ provided that such a representation does exist. The decisional bounded representation problem (DBR) is a variant of CR in which a bound $$c$$ and the factorization of $$\det A_f$$ are given as supplementary input and it is asked to decide the existence of primitive representations bounded by $$c$$. The decisional bounded equivalence problem (DBE) is a similar variant of CEP.
It is proved in the paper that the restriction of DBR to indefinite primitive forms and $$n=3$$, and the restriction to positive definite forms with $$n\geq 5$$, are NP-hard under probabilistic reductions. Also it is proved that the restriction of DBE to indefinite forms in which some bounding conditions are posed on parts of the matrix $$A_f$$ is NP-hard under probabilistic reductions.
The hardness of the above problems is used in order to build identification protocols. A prover should convince a verifier that he knows the solution $$T$$ of a given instance $$\text{CEP}(f,g)$$. The proposed protocol is correct and it is reasonably conjectured that the prover does not disclose the value of $$T$$. The protocols involve the calculation of LLL-reductions of quadratic forms.
##### MSC:
 11E20 General ternary and quaternary quadratic forms; forms of more than two variables 94A60 Cryptography 68W30 Symbolic computation and algebraic computation 11D09 Quadratic and bilinear Diophantine equations 11R29 Class numbers, class groups, discriminants
NTRUSign
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