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arXiv:0908.0022

Efficient quantum processing of ideals in finite rings

Paweł Wocjan, Stephen P. Jordan, Ahmadi, Hamed, Joseph P. Brennan

Published Jul 31, 2009• 0 citations
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Abstract

Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a quantum algorithm of Arvind et al. which finds a basis representation for R itself. We then show that our algorithm is a useful primitive allowing quantum computers to rapidly solve a wide variety of problems regarding finite rings. In particular we show how to test whether two ideals are identical, find their intersection, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit, and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. These problems appear to be hard classically.

DOI
10.48550/arxiv.0908.0022

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Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a quantum algorithm of Arvind et al. which finds a basis representation for R itself. We then show that our algorithm is a useful primitive allowing quantum computers to rapidly solve a wide variety of problems regarding finite rings. In particular we show how to test whether two ideals are identical, find their intersection, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit, and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. These problems appear to be hard classically.
Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.
Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring R=∠r1,...,rkR=\angle{r_1,...,r_k}R=∠r1​,...,rk​ where {r1,r2,...,rk}\{r_1,r_2,...,r_k\}{r1​,r2​,...,rk​} is an additive generating set for RRR and a multilinear polynomial f(x1,...,xm)f(x_1,...,x_m)f(x1​,...,xm​) over RRR also accessed as a black-box function f:Rm→Rf:R^m\to Rf:Rm→R (where we allow the indeterminates x1,...,xmx_1,...,x_mx1​,...,xm​ to be commuting or noncommuting), we study the problem of testing if fff is an \emph{identity} for the ring RRR. More precisely, the problem is to test if f(a1,a2,...,am)=0f(a_1,a_2,...,a_m)=0f(a1​,a2​,...,am​)=0 for all ai∈Ra_i\in Rai​∈R. We give a quantum algorithm with query complexity O(m(1+α)m/2kmm+1)O(m(1+α)^{m/2} k^{\frac{m}{m+1}})O(m(1+α)m/2km+1m​) assuming k≥(1+1/α)m+1k\geq (1+1/α)^{m+1}k≥(1+1/α)m+1. Towards a lower bound, we also discuss a reduction from a version of mmm-collision to this problem. We also observe a randomized test with query complexity 4mmk4^mmk4mmk and constant success probability and a deterministic test with kmk^mkm query complexity.
We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we also give evidence that this problem is hard for classical algorithms. The workings of the quantum algorithm rely on the interaction between the additive characters of the Fourier transform and the multiplicative characters of the Gauss sum.
Abstract A family of ring-based cryptosystems, including the multilinear maps of Garg, Gentry and Halevi [Candidate multilinear maps from ideal lattices, Advances in Cryptology—EUROCRYPT 2013, Lecture Notes in Comput. Sci. 7881, Springer, Heidelberg 2013, 1–17] and the fully homomorphic encryption scheme of Smart and Vercauteren [Fully homomorphic encryption with relatively small key and ciphertext sizes, Public Key Cryptography—PKC 2010, Lecture Notes in Comput. Sci. 6056, Springer, Berlin 2010, 420–443], are based on the hardness of finding a short generator of a principal ideal (short-PIP) in a number field typically in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} . In this paper, we present a polynomial-time quantum algorithm for recovering a generator of a principal ideal in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} , and we recall how this can be used to attack the schemes relying on the short-PIP in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} by using the work of Cramer et al. [R. Cramer, L. Ducas, C. Peikert and O. Regev, Recovering short generators of principal ideals in cyclotomic rings, IACR Cryptology ePrint Archive 2015, https://eprint.iacr.org/2015/313 ], which is derived from observations of Campbell, Groves and Shepherd [SOLILOQUY, a cautionary tale]. We put this attack into perspective by reviewing earlier attempts at providing an efficient quantum algorithm for solving the PIP in ℚ ⁢ ( ζ 2 s ) {\mathbb{Q}(\zeta_{2^{s}})} . The assumption that short-PIP is hard was challenged by Campbell, Groves and Shepherd. They proposed an approach for solving short-PIP that proceeds in two steps: first they sketched a quantum algorithm for finding an arbitrary generator (not necessarily short) of the input principal ideal. Then they suggested that it is feasible to compute a short generator efficiently from the generator in step 1. Cramer et al. validated step 2 of the approach by giving a detailed analysis. In this paper, we focus on step 1, and we show that step 1 can run in quantum polynomial time if we use an algorithm for the continuous hidden subgroup problem (HSP) due to Eisenträger et al. [K. Eisenträger, S. Hallgren, A. Kitaev and F. Song, A quantum algorithm for computing the unit group of an arbitrary degree number field, Proceedings of the 2014 ACM Symposium on Theory of Computing—STOC’14, ACM, New York 2014, 293–302].
Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently preformed on a quantum computer. The properties of the QFT over Galois rings lead to the quantum algorithm for hidden linear structures over Galois rings.