A Degree Bound For The c-Boomerang Uniformity Of Permutation Monomials

Let $\mathbb{F}_q$ be a finite field of characteristic $p$. In this paper we prove that the $c$-Boomerang Uniformity, $c \neq 0$, for all permutation monomials $x^d$, where $d > 1$ and $p \nmid d$, is bounded by $d^2$. Further, we utilize this bound to estimate the $c$-boomerang uniformity of a large class of Generalized Triangular Dynamical Systems, a polynomial-based approach to describe cryptographic permutations, including the well-known Substitution-Permutation Network

Controlling Nonequilibrium Phonon Populations in Single-Walled Carbon Nanotubes

We studied spatially isolated single-walled carbon nanotubes (SWNTs) immobilized in a quasi-planar optical λ/2-microresonator using confocal microscopy and spectroscopy. The modified photonic mode density within the resonator is used to selectively enhance or inhibit different Raman transitions of SWNTs. Experimental spectra are presented that exhibit single Raman bands only. Calculations of the relative change in the Raman scattering cross sections underline the potential of our microresonator for the optical control of nonequilibrium phonon populations in SWNT

Controlling molecular broadband-emission by optical confinement

We investigate experimentally and theoretically the fluorescence emitted by molecular ensembles as well as spatially isolated, single molecules of an organic dye immobilized in a quasi-planar optical microresonator at room temperature. The optically excited dipole emitters couple simultaneously to on- and off-axis cavity resonances of the microresonator. The multi-spectral radiative contributions are strongly modified with respect to free (non-confined) space due to enhancement and inhibition of the molecular spontaneous emission (SpE) rate. By varying the mirror spacing of the microresonator on the nanometer-scale, the SpE rate of the cavity-confined molecules and, consequently, the spectral line width of the microresonator-controlled broadband fluorescence can be tuned by up to one order of magnitude. Stepwise reducing the optical confinement, we observe that the microresonator-controlled molecular fluorescence line shape converges towards the measured fluorescence line shape in free space. Our results are important for research on and application of broadband emitters in nano-optics and -photonics as well as microcavity-enhanced (single molecule) spectroscopy

Arion: Arithmetization-Oriented Permutation and Hashing from Generalized Triangular Dynamical Systems

In this paper we propose the (keyed) permutation Arion and the hash function ArionHash over $\mathbb{F}_p$ for odd and particularly large primes. The design of Arion is based on the newly introduced Generalized Triangular Dynamical System (GTDS), which provides a new algebraic framework for constructing (keyed) permutation using polynomials over a finite field. At round level Arion is the first design which is instantiated using the new GTDS. We provide extensive security analysis of our construction including algebraic cryptanalysis (e.g. interpolation and Groebner basis attacks) that are particularly decisive in assessing the security of permutations and hash functions over $\mathbb{F}_p$. From a application perspective, ArionHash is aimed for efficient implementation in zkSNARK protocols and Zero-Knowledge proof systems. For this purpose, we exploit that CCZ-equivalence of graphs can lead to a more efficient implementation of Arithmetization-Oriented primitives. We compare the efficiency of ArionHash in R1CS and Plonk settings with other hash functions such as Poseidon, Anemoi and Griffin. For demonstrating the practical efficiency of ArionHash we implemented it with the zkSNARK libraries libsnark and Dusk Network Plonk. Our result shows that ArionHash is significantly faster than Poseidon - a hash function designed for zero-knowledge proof systems. We also found that an aggressive version of ArionHash is considerably faster than Anemoi and Griffin in a practical zkSNARK setting

Solving Degree Bounds for Iterated Polynomial Systems

For Arithmetization-Oriented ciphers and hash functions Gröbner basis attacks are generally considered as the most competitive attack vector. Unfortunately, the complexity of Gröbner basis algorithms is only understood for special cases, and it is needless to say that these cases do not apply to most cryptographic polynomial systems. Therefore, cryptographers have to resort to experiments, extrapolations and hypotheses to assess the security of their designs. One established measure to quantify the complexity of linear algebra-based Gröbner basis algorithms is the so-called solving degree. Caminata & Gorla revealed that under a certain genericity condition on a polynomial system the solving degree is always upper bounded by the Castelnuovo-Mumford regularity and henceforth by the Macaulay bound, which only takes the degrees and number of variables of the input polynomials into account. In this paper we extend their framework to iterated polynomial systems, the standard polynomial model for symmetric ciphers and hash functions. In particular, we prove solving degree bounds for various attacks on MiMC, Feistel-MiMC, Feistel-MiMC-Hash, Hades and GMiMC. Our bounds fall in line with the hypothesized complexity of Gröbner basis attacks on these designs, and to the best of our knowledge this is the first time that a mathematical proof for these complexities is provided. Moreover, by studying polynomials with degree falls we can prove lower bounds on the Castelnuovo-Mumford regularity for attacks on MiMC, Feistel-MiMC and Feistel-MiMCHash provided that only a few solutions of the corresponding iterated polynomial system originate from the base field. Hence, regularity-based solving degree estimations can never surpass a certain threshold, a desirable property for cryptographic polynomial systems

The Complexity of Algebraic Algorithms for LWE

Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gröbner basis computations on LWE polynomial systems under the assumption of semi-regularity. In this paper we revisit the Arora-Ge polynomial and prove that it satisfies a genericity condition recently introduced by Caminata & Gorla, called being in generic coordinates. For polynomial systems in generic coordinates one can always estimate the complexity of DRL Gröbner basis computations in terms of the Castelnuovo-Mumford regularity and henceforth also via the Macaulay bound. Moreover, we generalize the Gröbner basis algorithm of Semaev & Tenti to arbitrary polynomial systems with a finite degree of regularity. In particular, existence of this algorithm yields another approach to estimate the complexity of DRL Gröbner basis computations in terms of the degree of regularity. In practice, the degree of regularity of LWE polynomial systems is not known, though one can always estimate the lowest achievable degree of regularity. Consequently, from a designer\u27s worst case perspective this approach yields sub-exponential complexity estimates for general, binary secret and binary error LWE. In recent works by Dachman-Soled et al. the hardness of LWE in the presence of side information was analyzed. Utilizing their framework we discuss how hints can be incorporated into LWE polynomial systems and how they affect the complexity of Gröbner basis computations

A Zero-Dimensional Gröbner Basis for Poseidon

In this paper we construct dedicated weight orders $>$ so that a $>$-Gröbner bases of Poseidon can be found via linear transformations for the preimage as well as the CICO problem. In particular, with our Gröbner bases we can exactly compute the $\mathbb{F}_q$-vector space dimension of the quotient space for all possible Poseidon configurations. This in turn resolves previous attempts to assess the security of Poseidon against Gröbner basis attacks, since the vector space dimension quantifies the complexity of computing the variety of a zero-dimensional polynomial system

Exponential Decay Lifetimes of Excitons in Individual Single-Walled Carbon Nanotubes

The dynamics of excitons in individual semiconducting single-walled carbon nanotubes was studied using time-resolved photoluminescence (PL) spectroscopy. The PL decay from tubes of the same (n,m) type was found to be monoexponential, however, with lifetimes varying between less than 20 and 200 ps from tube to tube. Competition of nonradiative decay of excitons is facilitated by a thermally activated process, most likely a transition to a low-lying optically inactive trap state that is promoted by a low-frequency phonon mode