Abstract
Recently, Lyubashevsky & Seiler (Eurocrypt 2018) showed that small polynomials in the cyclotomic ring \(\mathbb {Z}_q[X]/(X^n+1)\), where n is a power of two, are invertible under special congruence conditions on prime modulus q. This result has been used to prove certain security properties of lattice-based constructions against unbounded adversaries. Unfortunately, due to the special conditions, working over the corresponding cyclotomic ring does not allow for efficient use of the Number Theoretic Transform (NTT) algorithm for fast multiplication of polynomials and hence, the schemes become less practical.
In this paper, we present how to overcome this limitation by analysing zeroes in the Chinese Remainder (or NTT) representation of small polynomials. As a result, we provide upper bounds on the probabilities related to the (non)-existence of a short vector in a random module lattice with no assumptions on the prime modulus. We apply our results, along with the generic framework by Kiltz et al. (Eurocrypt 2018), to a number of lattice-based Fiat-Shamir signatures so they can both enjoy tight security in the quantum random oracle model and support fast multiplication algorithms (at the cost of slightly larger public keys and signatures), such as the Bai-Galbraith signature scheme (CT-RSA 2014), \(\mathsf {Dilithium\text {-}QROM}\) (Kiltz et al., Eurocrypt 2018) and \(\mathsf {qTESLA}\) (Alkim et al., PQCrypto 2017). Our techniques can also be applied to prove that recent commitment schemes by Baum et al. (SCN 2018) are statistically binding with no additional assumptions on q.
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- 1.
Lyubashevsky and Seiler [19] showed, however, how to combine the FFT algorithm and Karatsuba multiplication in order to multiply in partially-splitting rings faster.
- 2.
Alternatively, we call it “FFT/NTT representation” in the fully-splitting case.
- 3.
This technique has already been investigated in the literature for e.g. constructing provably secure variants of NTRUEncrypt [25].
- 4.
We present it in the full version of this paper [20].
- 5.
What we mean by “small” is that the polynomial has small infinity or Euclidean norm.
- 6.
Namely, for each \(a_i\) we define a corresponding column vector \((a'_{i,1},...,a'_{i,d})\), where \(a'_{i,j}\) is the element of \(\mathbb {Z}_q[X]/ (f_j(X))\), such that \(a_i \equiv a'_{i,j} \ (\mathrm {mod}\ f_j(X))\), for \(j \in [d]\).
- 7.
In the example above, \(W_1\) is represented by the set \(\{X^j : j \in [2n]\}\). Indeed, \(|\mathsf {Zero}(X^j - X^k)| < 1\) for all distinct j, k.
- 8.
Note that this technique can also be used for \(W_1\) as long as \(q^{1/d}\) is large enough.
- 9.
This can be proven similarly as in [5] by putting a box of side-length 1 centered on every integer point and checking that the ball is completely covered by these boxes.
- 10.
For readers not familiar with definitions of lossy and canonical identification schemes, we provide all necessary background in [20].
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Acknowledgments
The author would like to thank Vadim Lyubashevsky for fruitful discussions and anonymous reviewers for their useful comments. This work was supported by the SNSF ERC Transfer Grant CRETP2-166734 FELICITY.
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Nguyen, N.K. (2019). On the Non-existence of Short Vectors in Random Module Lattices. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11922. Springer, Cham. https://doi.org/10.1007/978-3-030-34621-8_5
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