figure table

# Ground test results of the micro-vibration interference for the x-ray
microcalorimeter onboard XRISM

Takashi Hasebe  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo, Chiba, Japan  Ryuta Imamura  Graduate
School of Science and Engineering, Ehime University, Ehime, Japan  Masahiro
Tsujimoto  Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency (JAXA), Kanagawa, Japan  Hisamitsu Awaki  Graduate School
of Science and Engineering, Ehime University, Ehime, Japan  Meng P. Chiao
National Aeronautics and Space Administration (NASA), Goddard Space Flight
Center, MD, USA  Ryuichi Fujimoto  Faculty of Mathematics and Physics,
Kanazawa University, Ishikawa, Japan  Leslie S. Hartz  National Aeronautics
and Space Administration (NASA), Goddard Space Flight Center, MD, USA
Caroline A. Kilbourne  National Aeronautics and Space Administration (NASA),
Goddard Space Flight Center, MD, USA  Gary A. Sneiderman  National Aeronautics
and Space Administration (NASA), Goddard Space Flight Center, MD, USA  Yoh
Takei  Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency (JAXA), Kanagawa, Japan  Susumu Yasuda  Japan Aerospace
Exploration Agency (JAXA), Ibaraki, Japan  the XRISM Resolve team

###### Abstract

Resolve is a payload hosting an X-ray microcalorimeter detector operated at 50
mK in the X-Ray Imaging and Spectroscopy Mission (XRISM). It is currently
under development as part of an international collaboration and is planned to
be launched in 2023\. A primary technical concern is the micro-vibration
interference in the sensitive microcalorimeter detector caused by the
spacecraft bus components. We conducted a series of verification tests in
2021–2022 on the ground, the results of which are reported here. We defined
the micro-vibration interface between the spacecraft and the Resolve
instrument. In the instrument-level test, the flight-model hardware was tested
against the interface level by injecting it with micro-vibrations and
evaluating the instrument response using the 50 mK stage temperature
stability, ADR magnet current consumption rate, and detector noise spectra. We
found strong responses when injecting micro-vibration at $\sim$200, 380, and
610 Hz. In the former two cases, the beat between the injected frequency and
cryocooler frequency harmonics were observed in the detector noise spectra. In
the spacecraft-level test, the acceleration and instrument responses were
measured with and without suspension of the entire spacecraft. The reaction
wheels (RWs) and inertial reference units (IRUs), two major sources of micro-
vibration among the bus components, were operated. In conclusion, the observed
responses of Resolve are within the acceptable levels in the nominal
operational range of the RWs and IRUs. There is no evidence that the resultant
energy resolution degradation is beyond the current allocation of noise
budget.

###### keywords:

low temperature detector, x-ray microcalorimeter, micro-vibration
interference, XRISM

*Takashi Hasebe<EMAIL_ADDRESS>
##  1 Introduction

The X-Ray Imaging and Spectroscopy Mission (XRISM) is an x-ray observatory
planned to be launched in 2023 [1]. One of the scientific payloads is
Resolve[2], which hosts an x-ray microcalorimeter for high resolution
spectroscopy. It was developed under the collaboration of JAXA, NASA, ESA, and
universities and institutes under these agencies. Resolve is based on the soft
x-ray spectrometer (SXS)[3, 4] onboard ASTRO-H[5] that was launched in 2016
February. The SXS achieved a stable in-orbit spectroscopic performance with an
energy resolution of 5 eV FWHM at 5.9 keV[6]. The success was suddenly
terminated by the loss of the spacecraft attitude control in a month. Resolve
was made with almost the same design as the SXS to recover its scientific
programs.

Micro-vibration interference is a primary concern in the design of instruments
hosting a low-temperature detector, which is sensitive to all forms of energy
inputs dissipating into heat. In particular, microcalorimeters or bolometers
that use conventional high-impedance semiconductor detectors coupled with
mechanical cryocoolers in the cooling chain are susceptible to this
interference. Examples include the Planck[7] HFI[8] using neutron-transmuted
doped Ge of $\sim$10 M$\Omega$ operated at 100 mK, and the ASTRO-H SXS and
XRISM Resolve using ion-implanted Si of $\sim$30 M$\Omega$ at 50 mK. In fact,
strong pickup lines were observed in the detector noise spectra both in the
HFI and SXS at the driving frequency of the cryocoolers (15–52 Hz) and their
harmonics.

In SXS, the cryocooler micro-vibration degraded detector performance to an
unacceptable level. The temperature fluctuation at the 50 mK stage was
$\sim$40 $\mu$K rms, whereas 2.5 $\mu$K was budgeted. As a countermeasure, a
significant change was made in the hardware at a very late stage of
development. The vibration isolation system (VIS) was newly designed and
installed between the cryocooler compressors and cryostat a few months before
the final integration into the spacecraft[9]. This countermeasure was
successful and reduced the fluctuation to 0.5 $\mu$K rms. For Resolve, the VIS
between the cryocoolers and the cryostat is included from the beginning. It
has an improved design that includes a launch-lock mechanism and has better
dumping performance[10]. The micro-vibration interference by the cryocoolers
to the detector is discussed in a separate paper[11, 12].

In addition to the cryocoolers in the instrument, the spacecraft also hosts
potential micro-vibration sources, such as reaction wheels (RW) and inertial
reference units (IRU) in the attitude control system. In the case of SXS,
these components exhibited some hints of micro-vibration interference during
the spacecraft test. However, the testing time on the ground and the life in
orbit were insufficient to assess their impact and isolate the problem.
Spacecraft interference poses a challenge in the integration program as it can
be identified only after the instrument is integrated into the spacecraft when
limited resources remain for major changes. For Resolve, a micro-vibration
control plan was formulated and a series of tests were performed to control
the risk. The purpose of this article is to describe the results obtained in
the series of ground tests from 2021 to 2022 at JAXA’s Tsukuba Space Center
using flight model hardware. We aim to provide a case study of the micro-
vibration control of a space cryogenic mission, which will be followed by many
others.

The remainder of this paper is organized as follows. In § 2, we give a brief
description of the Resolve instrument and the spacecraft bus system, and the
micro-vibration interface between them. The sensitivity of Resolve to micro-
vibration was tested before integration onto the spacecraft through the
injection of micro-vibrations at simulated interfaces to the spacecraft (§ 3).
The actual level of micro-vibration at the interface by the spacecraft was
measured and an end-to-end assessment from the bus components to the detector
was made in the spacecraft-level test (§ 4). We summarize the result in § 5.

##  2 Hardware and Interface

###  2.1 Spacecraft

The spacecraft design of XRISM is based on ASTRO-H’s. NEC is the prime
contractor for both. The spacecraft will be put into a near-earth orbit at an
altitude of 575 $\pm$ 15 km and an inclination angle of 31 degrees. A
schematic view is shown in Figure 1 left. It has a size of $7.9\times
9.2\times 3.1$ m3 and a weight of $2.3\times 10^{3}$ kg[1]. The main body is
made of eight side panels (SP1–8) of $990\times 3100$ mm2 and the base, lower,
middle, and top plates perpendicular to the side panels. X-ray telescopes and
the star trackers are placed on the top plate, while Resolve dewar is on the
base plate. The solar panel is attached in parallel with the SP3 toward the
Sun, while the dewar is exposed to the deep space through the open space on
the opposite side (SP7) for radiative cooling.

Figure 1: Schematic view of the spacecraft (left) and cross-sectional view
inside, in which the lower and middle plates and struts are removed (right).
The drawing is provided by NEC.

The spacecraft has four RWs (RW1–4) to maintain the pointing during
observations and three magnetic torquers to dump the accumulated angular
momentum against the Earth’s magnetic field. The attitude is measured by four
independent sensors: IRUs, the geomagnetic aspect sensors, the digital sun
sensors, and the star trackers. These actuators, sensors, and their driving
electronics are stored inside the side panels (Figure 1 right). The RWs and
IRUs are the two major sources of micro-vibration in the spacecraft system.
The RWs are operated for the three-axis control at a nominal rotation speed of
1.5–4.5 krpm (25–75 Hz) with the base at 3.0 krpm (50 Hz). Two (RW1 and 3) are
located inside of the SP2 and the other two (RW2 and 4) in SP4. The IRUs
consist of two sets (IRU-A and B) of two tuned dry gyros, and three of them
are used at a fixed rotation speed of 155 Hz. The IRUs are stored in the
spacecraft base panel to avoid thermal distortion of the side panels. These
frequencies are right in the middle of the microcalorimeter bandpass of 12 to
a few hundred Hz.

###  2.2 Instrument

The x-ray microcalorimeter detector has an array of 6$\times$6 pixels that are
thermally anchored to the 50 mK bath with a thermal time constant of a few
ms[6]. The detector is placed inside the dewar, which consists of several
layers of shields made of Al for thermal insulation and vacuum[13, 14]. Inside
the dewar, the superfluid He is stored in the He tank. Two-stage adiabatic
demagnetization refrigerator (ADR) is used to pump heat from the 50 mK stage
to the He tank[15]. The evaporated He is used to vapor-cool the shields inside
the dewar. When the He runs out, a third ADR is used to cool down the He tank
to maintain the 50 mK control for an extended lifetime[16, 17]. On the surface
of the dewar, five cryocoolers are attached for active cooling of the shields
to reduce the thermal load toward the coldest stage. Two Stirling coolers
(STC)[18] cool the 100 and 30 K stages, while one Joule-Thomson cooler (JTC)
cool the 4 K stage. The other two STCs are used for pre-cooling the JTC. The
STC’s have a compressor containing two opposing pistons and are driven at 15
Hz, while the JTC has two compressors containing two opposing pistons to
generate low and high pressures and is driven at 52 Hz. The frequency can be
tuned by $\pm$1 Hz for STC and $\pm$2 Hz for JTC by commands to avoid possible
resonances. Each pixel of the detector is read out independently at a sampling
rate of 12.5 kHz. The readout electronics called the XBOX[3] supplies the
detector bias, and samples, shapes, amplifies, and digitizes the detector
signal. The ADR controller (ADRC[3]) controls the ADR magnet current and the
heat switches as well as the thermal control of the cold stages inside the
dewar. These two room-temperature electronics have wires reaching deep inside
the dewar, thus are considered to be possible routes of micro-vibration into
the detector. Both are placed inside of the SP1 (Figure 1 right).

###  2.3 Interface

We defined the micro-vibration interface between the spacecraft and the
instrument at three positions (Figure 2): (1) The spacecraft base plate, on
which the dewar stands with its eight struts. They are the only points that
the dewar is mechanically supported on the spacecraft. (2) The SP1, at the
foot of XBOX, where the XBOX and ADRC are located. (3) The harness connecting
the dewar and the XBOX and ADRC. The harness is supported by the brackets on
the base plate, which are defined as an interface. For each interface, the
maximum allowed vibration input as a function of frequency was estimated based
on the allocation for the micro-vibration interference in the Resolve noise
budget. This limit was then imposed as a requirement on the bus components.
The levels are defined in acceleration (m s-2) as a function of frequency in
the $z$ axis for (1) and (3) and $x$ axis for (2) perpendicular to SP1. The
interface levels and frequency ranges (Figure 4) were given based on the SXS
results for (1) and on a component-level test for (2). The stricter of the two
was applied for (3).

Figure 2: Schematic view of the spacecraft–Resolve interfaces and the photos
of vibrators used in the instrument-level test. The drawing is provided by
NEC.

##  3 Instrument-level tests

###  3.1 Measurements

We conducted the instrument-level test in 2021 December 4–11. The Resolve
instrument was operated in the nominal configuration. The drive frequencies of
the STC and JTC were 14.452 and 53.478 Hz, respectively. We fabricated the
spacecraft interface simulators for the base plate, the harness support
brackets, and the SP1, where the interfaces between the spacecraft and the
instrument are defined. Note that there are some differences; the base plate
and the SP1 simulators are made of Al, while the flight equivalent units are
of Al honeycomb with the CFRP skin. At each interface, we attached a vibrator
with a force sensor for injecting micro-vibration (Figure 2) and piezoelectric
accelerometers for monitoring levels (Figure 3). The accelerometers were read
out continuously at a 1.6 kHz sampling rate with a low pass filter cut off at
$\sim$800 Hz. On the base plate simulator, a Wilcoxson F4 vibrator was placed
at a position displaced from the center in the $+y$ direction and four
accelerometers 90 degrees apart (DWR BP $\pm x\\_Z$ or $\pm y\\_Z$) were
placed along the $z$ axis. On the back side of the SP1 simulator, an F3
vibrator and two sets of three-axis accelerometers were attached. Close to the
bracket, another F3 vibrator and two other sets of accelerometers for the
three axes were placed.

Figure 3: Positions of the accelerometers and their identifications. The XBOX1
and XBOX2 accelerometers are placed on the backside of the SP1 simulator. The
drawing is provided by NEC.

For each interface, we conducted three runs: (i) force calibration, (ii)
frequency sweep, and (iii) dwell. For (i), both the force and accelerometer
sensors were read by giving a fixed force over the interface frequency range,
from which the force necessary to achieve the interface level of the
acceleration was derived as a function of frequency. For (ii), we swept over
the frequency at the derived force (Figure 4) and evaluated the excess heat
load to the detector and its stability using the ADR magnet current
consumption rate ($dI_{\mathrm{ADR}}/dt$) and the 50 mK temperature
fluctuation ($T_{\mathrm{rms}}$) sampled at 1 Hz. Two neutron transmutation
doped Germanium resistance thermometers are used for the 50 mK stage; one is
for the PID control of the magnet current and the other for monitor. These
thermometers are known to exhibit glitches in reading, thus we judged the
positive response only when at least two of the three measurements (two
$T_{\mathrm{rms}}$ and one $dI_{\mathrm{ADR}}/dt$ values) agreed. We tried
injecting the interface level or beyond as accurately as possible, but ended
up injecting lower levels in some frequencies, in particular in the low-
frequency range, due to the power limitation of the vibrators. For such
frequencies, we made additional dwell measurements for linearity check. For
(iii), we dwell at selected frequencies based on the sweep results for 5 min
to collect the detector noise spectra. We also changed the amplitudes of the
micro-vibration injection to examine the linearity of the response.

(a) Interface (1) Base plate (b) Interface (2) Side panel 1 (c) Interface (3)
Harness support bracket

Figure 4: Interface levels and the injected levels during the frequency sweep.

###  3.2 Results

We first examine the $T_{\mathrm{rms}}$ values against the acceptable level of
2.5 $\mu$K rms for the three interface injections. No deviation was found for
the interface 2 (SP1). The result of the interface 3 (harness support bracket)
was encompassed in that of the interface 1 (base plate). We thus focus on the
result of the base plate injection. Figure 5 shows the $T_{\mathrm{rms}}$ and
$dI_{\mathrm{ADR}}/dt$ responses as a function of the injection frequency. The
positive responses were found at $\sim$200, 380, and 610 Hz. The one at
$\sim$200 Hz is far beyond the acceptable level of $T_{\mathrm{rms}}$.

Figure 5: Stability of the 50 mK stage thermometers ($T_{\rm{rms}}$; top) and
the ADR magnet current consumption rate ($dI_{\mathrm{ADR}}/dt$; bottom)
during the frequency sweep injection from the base plate. The horizontal
dashed line in the top panel indicates the acceptable level of 2.5 $\mu$K rms.
The result was obtained by injecting a level higher than that defined in
figure 4 (a) to investigate possible margins.

The former two frequencies are of particular interest as they match with the
4’th and 7’th harmonics of the JTC drive frequency, at which peculiar
responses in the detector noise spectra were observed during the JTC frequency
tuning in different experiments[11, 12]. We did not observe such responses in
other JTC harmonics. The peculiar response is the presence of the low-
frequency beat between 4’th harmonics of the JTC and 14 or 15’th harmonics of
the STC for $\sim$200 Hz and between 7’th of the JTC and 24’th or 25’th of the
STC for $\sim$380 Hz. We speculate that the instrument has a resonance at
these frequencies in cold stages, where the two harmonic lines of the JTC and
STC are mixed and amplified non-linearly to redistribute some power into the
beat frequency at $<$20 Hz. They were found to be dissipated somehow into heat
at the cold stage, as the beat frequency noise was not observed when the
detector bias was off to be insensitive to heat[11, 12]. The elevated levels
of $T_{\rm{rms}}$ and $dI_{\mathrm{ADR}}/dt$ is considered to be related to
these resonances.

The low-frequency lines such as this degrade the detector performance
significantly. Before this test took place, we tuned the JTC and STC
frequencies so that they do not cause a low-frequency beat. However, during
the micro-vibration test, a third line was injected at these resonant
frequencies, which was mixed with the existing JTC and STC harmonic lines. We
obtained the detector noise spectra during the micro-vibration injection of
these frequencies (Figure 6). The low-frequency lines are evident, which are
attributable to the beats among the injected frequency and the existing JTC
and STC harmonics. In the 200 Hz injection (left), the line at 2.2 Hz is the
beat between the injected frequency and the 14’th STC harmonic, and the one at
13.9 Hz is between the injected frequency and the 4’th JTC harmonic. In the
380.4 Hz injection (right), the line at 1.4 Hz is the beat between the 7’th
JTC and 24’th STC harmonic and the one at 6.1 Hz is between the injected
frequency and the 7’th JTC harmonic.

Figure 6: Noise spectrum of the pixel 0 of the microcalorimeter. (left) 200 Hz
injection from the harness support bracket and (right) 380.4 Hz injection from
the base plate. The black vertical line indicates the frequency of injection,
while the blue and red lines are the beat frequencies among the injected
frequency and STC or JTC harmonic frequencies.

We are now concerned that the spacecraft system may bring the third line at
the frequencies of the enhanced sensitivity, particularly at $\sim$200 Hz with
an unacceptable level of response. The drive frequencies of the RWs (25–75 Hz)
and the IRUs (155 Hz) are lower than this, but their non-diagonal
redistribution component into higher frequencies may hit the sensitive
frequencies. In the RW unit test, such a non-diagonal component was indeed
found. The coupling between the eigen frequency of the RW at rest
($\omega_{0}$) and the rotating frequency ($\Omega$) causes precession
frequencies at

$\omega_{p}^{(\pm)}=\pm\Omega+\sqrt{\left(\frac{I_{zz}\Omega}{2I_{rr}}\right)^{2}+\omega_{0}},$
(1)

in which $I_{zz}$ and $I_{rr}$ are the moment of inertia around the rotating
and radial axes of the RW, respectively[19]. The two signs are for the forward
and backward branches. When $\Omega=66$ Hz (3.96 krpm) within the operation
range, the forward branch ($\omega_{p}^{(+)}$) matches $\sim$200 Hz. The study
of the impact of this mode is left for the spacecraft-level test (§ 4).

##  4 Spacecraft-level tests

###  4.1 Measurements

We performed the spacecraft-level tests twice in 2022 May 23–25 and June 10–11
in different configurations. In the former, the entire spacecraft was
suspended by a crane for flight fidelity of the mechanical environment, but
the Resolve instrument was not operated for the safety of the He plumbing
(Figure 7 left). In the latter, the spacecraft was placed on the floor, but
the detector was operated at 50 mK to make an end-to-end assessment (Figure 7
right). The STC and JTC were operated at 14.631 and 51.588 Hz after retuning
the frequency prior to the spacecraft integration. The two tests are hereafter
called the suspended and unsuspended tests.

Figure 7: Spacecraft-level test configurations: (left) Spacecraft suspended
and no Resolve operation and (right) Spacecraft on the floor and Resolve
operation.

In both tests, a total of 18 accelerometers (3 axes at 6 positions) were used
to monitor relevant positions of the spacecraft base plate, harness support
bracket, SP1, and the outermost shell of the dewar (Figure 3). In the
unsuspended test, the detector noise spectra were additionally obtained on
demand.

In the suspended test, we performed a full set of operations of the RWs and
the IRUs. Each RW was run up to the maximum 6.0 krpm and coasted down to 0,
one by one, sweeping over its entire frequency range. Each IRU was run at 155
Hz one by one, and a combination of three out of the four for simulating the
flight use. In the unsuspended test, each RW was set at selected frequencies
(1.5, 3.0, 3.96, and 6.0 krpm) to collect the detector noise spectra. All RW
units were coasted down from 6.0 krpm to 0. The start was set apart by 30–60
min, so that their responses can be distinguished by the concurrent RW
rotating speeds. During the coast down, the detector noise spectra were
obtained on demand every 3 min to track the RW rotation changes within the
allocated data rate.

###  4.2 Results

First, we compare the accelerometer levels in the two configurations with and
without suspending the spacecraft. Figure 8 shows the power in the
accelerometer spectra at 100 and 155 Hz respectively during the 6 krpm RW
rotation and the IRU operation. In the unsuspended test, the cryocoolers
dominate the spectrum, but the RW and the IRU signatures are distinctive at
their fundamental frequencies. The power in the lines, from which the local
continuum level was subtracted, varies from one accelerometer channel to
another, but there is no systematic difference between the suspended and
unsuspended results. We can use the result of the unsuspended test for an end-
to-end assessment with sufficient flight fidelity.

Figure 8: Comparison of accelerometer power between the suspended (red) and
unsuspended (blue) tests for RW1 at 6.0 krpm and RW2–4 with no operation
(circles) and an IRU operation (crosses).

Next, we examine the accelerometer spectra of the IRUs and the RWs in the
suspended test, which is uncontaminated by the cryocooler lines, with a
particular focus on the non-diagonal components into high frequencies. Figure
9 shows the spectrum taken during the IRU operation compared to the
background. The fundamental line at 155 Hz and its second harmonic (but not
third and beyond) were observed. The acceleration at 155 Hz is below the
interface level of the $1.4\times 10^{-3}$ m s-2. Figure 10 shows the
accelerometer spectrogram as a function of the RW rotation speed when the RW1
was coasted down. The line noise caused by the RW should leave a trace that
varies with time, or the concurrent RW rotation speed, while the background
line noise should leave a horizontal trace constant with time. In fact, the
constant background was removed in the plot by subtracting the median of all
spectra taken during the coast down. Besides the diagonal ($x=y$) traces by
the fundamental frequency, a dozen non-diagonal redistribution traces are
recognized by the higher order harmonics. There is no clear signature of the
precession lines that would leave curved traces in this representation. By
taking the strongest signal at each accelerometer frequency throughout the
coast down, we can assess the worst level. The result was below the interface
level in the operating range (1.5–4.5 krpm) of all the RWs (Figure 11).

Figure 9: Accelerometer spectrum of IRU in comparison to the background during
the suspended test for the HNS1-Z accelerometer. The fundamental frequency of
155 Hz and its harmonics are shown with dotted lines.

(a) Accelerometer

(b) Detector noise

Figure 10: Spectrogram of the accelerometer spectra in the suspended test
(left) and the detector noise spectra in the unsuspended test (right) during
the RW1 coast down. The horizontal axis is translated from time to the
concurrent RW rotation speed. The color scale is for the differential spectra
by subtracting the median along the horizontal axis for the stationary
component. Only positive signals are shown. The residual non-stationary
horizontal line at 100 Hz is due to the RW2–4 that kept rotating at 6 krpm
during RW1 coast down and that at 530 Hz is due to 10’th harmonic of JTC.
Figure 11: Strongest accelerometer spectra at each frequency measured during
the RW coast down when the rotation is in its nominal operational range
(1.5–4.5 krpm) for the dewer base plate (left), side panel 1 (center), and
harness support bracket (right) in the suspended test for channels
representing the interface axis. The dashed lines indicate the interface
levels.

We now explore the response of the Resolve instrument using the data taken in
the unsuspended test. We examined the 50 mK stage temperature fluctuation
($T_{\mathrm{rms}}$) and the ADR magnet current consumption rate
($dI_{\mathrm{ADR}}/dt$) during the coast down. Figure 12 shows the result
during the RW1 coast down. A positive response was found around 16 Hz, which
was unique to RW1. The RW1 response at $\sim$16 Hz was confirmed in the coast
downs during the spacecraft thermal-vacuum test held in 2022 August. No
positive response was found in the operating range of 25–75 Hz for all the RW
units. We thus conclude that the micro-vibration of the RWs does not cause a
serious thermal load to the 50 mK stage within its nominal operating range.

Figure 12: Same with Figure 5 during the RW1 coast down in the unsuspended
test (blue), thermal-vacuum (Tvac) test cold and hot cases (cyan and magenta
respectively). The spacing of the data is non-uniform along the RW, as the
rotation decelerates faster at faster rotations during the coast down.

We next check the detector noise spectra taken during the coast down. Figure
10 shows the spectrogram during the RW1 coast down using all 36
microcalorimeter pixels for the sake of statistics. The constant component was
subtracted in the same way as Figure 10. The fundamental line of the RW
rotation speed is clearly seen with a decreasing intensity toward decreasing
frequency, but no obvious non-diagonal lines were found. We compared the power
of the diagonal component for each unit of RW1–4 during their coast down in
Figure 13. We found that the RW1 shows the largest response.

Finally, we make an end-to-end assessment in terms of the detector energy
resolution. In one of the RW coast down tests during the spacecraft thermal
vacuum test, we illuminated 55Fe X-ray sources over the entire detector array
to take the X-ray spectra. Being limited by the statistics, we compared the
energy resolution during the former and latter halves of the coast down and
that before the coast down. We found no significant changes in the Mn
K$\alpha$ energy resolution among them. We also performed a long ($>$8 hr)
55Fe X-ray integration several times by operating the RW1–4 at a fixed rate of
3 kprm and the magnetic torqer, which is a known electromagnetic interference
source[20], at some fixed duty ratio. Comparison with a reference measurement
of no operation of these interference sources showed no significant difference
in the performance of individual pixels. When the data of 33 pixels were
combined, a barely significant difference of $<$0.02 eV for the Mn K$\alpha$
energy resolution of $\sim$4.0 eV.

Figure 13: Line power of the detector noise spectra at the frequency of the
concurrent rotating speed of each RW unit during coast down in the unsuspended
test.

##  5 Summary

We presented the result of a series of ground tests to verify the micro-
vibration interference from the spacecraft bus system to the microcalorimeter
detector of Resolve onboard XRISM. In the instrument-level test, we injected
micro-vibration as a function of frequency with a force to yield the
spacecraft-instrument interface level. We evaluated the detector response
using the 50 mK stage temperature stability and the ADR magnet current
consumption rate. We found the strong responses at $\sim$ 200, 380, and 610
Hz. In the former two cases, the beat among the injected frequency and the
cryocooler frequency harmonics of the JTC and the STC are observed in the
detector noise spectra. In the spacecraft-level test, we operated the two
major sources of the micro-vibration in the spacecraft —the RWs and the IRUs—
with suspending the entire spacecraft for the mechanical flight fidelity and
without suspending the spacecraft and with operating Resolve for an end-to-end
assessment. We observed numerous noise lines besides the fundamental
frequencies of the RWs and IRUs, but none of them cause an unacceptable level
of response in the instrument within the nominal operational ranges of the RWs
and IRUs. We conclude that the verification of the micro-vibration is complete
and the risk is sufficiently mitigated for the flight.

It would have been difficult to assess the micro-vibration interference
outcome before we actually did an end-to-end test, especially when we have
non-diagonal injection and response as shown here. In case of XRISM, it was
fortunate that none of the micro-vibration sources in the spacecraft brought
noise at frequencies that the detector is sensitive to. If it were the case,
we had an optional solution to change the nominal frequency range of RWs. Such
flexibility would be worth considering in the design of future missions.

###### Acknowledgements.

This work is made possible only with significant contributions from all the
XRISM Resolve team members and the engineers of SHI and NEC.

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Takashi Hasebe is a post doctoral fellow of the Kavli Institute for the
Physics and Mathematics, the University of Tokyo. He received his Ph.D. degree
in physics from Hiroshima University in 2016. His current research interests
include cryogenics and millimeter-wave optics.

###### List of Figures

  1. 1 Schematic view of the spacecraft (left) and cross-sectional view inside, in which the lower and middle plates and struts are removed (right). The drawing is provided by NEC.
  2. 2 Schematic view of the spacecraft–Resolve interfaces and the photos of vibrators used in the instrument-level test. The drawing is provided by NEC.
  3. 3 Positions of the accelerometers and their identifications. The XBOX1 and XBOX2 accelerometers are placed on the backside of the SP1 simulator. The drawing is provided by NEC.
  4. 4 Interface levels and the injected levels during the frequency sweep.
  5. (a) Interface (1) Base plate
  6. (b) Interface (2) Side panel 1
  7. (c) Interface (3) Harness support bracket
  8. 5 Stability of the 50 mK stage thermometers ($T_{\rm{rms}}$; top) and the ADR magnet current consumption rate ($dI_{\mathrm{ADR}}/dt$; bottom) during the frequency sweep injection from the base plate. The horizontal dashed line in the top panel indicates the acceptable level of 2.5 $\mu$K rms. The result was obtained by injecting a level higher than that defined in figure 4 (a) to investigate possible margins.
  9. 6 Noise spectrum of the pixel 0 of the microcalorimeter. (left) 200 Hz injection from the harness support bracket and (right) 380.4 Hz injection from the base plate. The black vertical line indicates the frequency of injection, while the blue and red lines are the beat frequencies among the injected frequency and STC or JTC harmonic frequencies.
  10. 7 Spacecraft-level test configurations: (left) Spacecraft suspended and no Resolve operation and (right) Spacecraft on the floor and Resolve operation.
  11. 8 Comparison of accelerometer power between the suspended (red) and unsuspended (blue) tests for RW1 at 6.0 krpm and RW2–4 with no operation (circles) and an IRU operation (crosses).
  12. 9 Accelerometer spectrum of IRU in comparison to the background during the suspended test for the HNS1-Z accelerometer. The fundamental frequency of 155 Hz and its harmonics are shown with dotted lines.
  13. 10 Spectrogram of the accelerometer spectra in the suspended test (left) and the detector noise spectra in the unsuspended test (right) during the RW1 coast down. The horizontal axis is translated from time to the concurrent RW rotation speed. The color scale is for the differential spectra by subtracting the median along the horizontal axis for the stationary component. Only positive signals are shown. The residual non-stationary horizontal line at 100 Hz is due to the RW2–4 that kept rotating at 6 krpm during RW1 coast down and that at 530 Hz is due to 10’th harmonic of JTC.
  14. 11 Strongest accelerometer spectra at each frequency measured during the RW coast down when the rotation is in its nominal operational range (1.5–4.5 krpm) for the dewer base plate (left), side panel 1 (center), and harness support bracket (right) in the suspended test for channels representing the interface axis. The dashed lines indicate the interface levels.
  15. 12 Same with Figure 5 during the RW1 coast down in the unsuspended test (blue), thermal-vacuum (Tvac) test cold and hot cases (cyan and magenta respectively). The spacing of the data is non-uniform along the RW, as the rotation decelerates faster at faster rotations during the coast down.
  16. 13 Line power of the detector noise spectra at the frequency of the concurrent rotating speed of each RW unit during coast down in the unsuspended test.


y,\\\ 0&\mathrm{N}\mathfrak{ln}>y.\end{array}\right.$

Then it follows from (8.7) that

$\delta(\log\mathrm{N}\mathfrak{ln})=\frac{1}{\pi}\int_{-A}^{A}(\mathrm{N}\mathfrak{ln})^{i\alpha}\frac{\sin\gamma\alpha}{\alpha}d\alpha+O\Big{(}\frac{y}{A}\Big{)}.$

Hence

${\underset{\mathrm{N}\mathfrak{ln}\leqslant
y\atop\mathfrak{ln}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})b(\mathfrak{n})={\underset{\mathfrak{ln}\equiv\mathfrak{a}\text{
in }\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})b(\mathfrak{n})\delta(\log\mathrm{N}\mathfrak{ln})\\\
=\frac{1}{\pi}\int_{-A}^{A}{\underset{\mathfrak{ln}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})(\mathrm{N}\mathfrak{l})^{i\alpha}b(\mathfrak{n})(\mathrm{N}\mathfrak{n})^{i\alpha}\frac{\sin\gamma\alpha}{\alpha}d\alpha+O\Big{(}\frac{y}{A}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|a(\mathfrak{l})b(\mathfrak{n})|\Big{)}.$

The error term here is manageable if we take $A=xLN$. For $y\leqslant x$, the
integral is

$\ll\int_{-A}^{A}\Big{|}{\underset{\mathfrak{ln}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})(\mathrm{N}\mathfrak{l})^{i\alpha}b(\mathfrak{n})(\mathrm{N}\mathfrak{n})^{i\alpha}\Big{|}\min\Big{(}\log
x,\frac{1}{|\alpha|}\Big{)}d\alpha.$

Summing the integrand over modulus $\mathfrak{m}$ and applying Lemma 8.1, we
have

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}{\underset{\mathfrak{ln}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})(\mathrm{N}\mathfrak{l})^{i\alpha}b(\mathfrak{n})(\mathrm{N}\mathfrak{n})^{i\alpha}\Big{|}$
$\displaystyle\ll\Big{(}Q+\sqrt{L+N}+\frac{\sqrt{LN}}{(\log
N)^{A}}\Big{)}(\log
Q)^{2}\mathcal{A}(L)^{\frac{1}{2}}\mathcal{B}(N)^{\frac{1}{2}}.$

Since $\int_{-A}^{A}\min(\log x,|\alpha|^{-1})d\alpha\ll\log xLN$ trivially,
the lemma follows. ∎

##  9\. Proof of Theorem 3.1

It remains to estimate the averages

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{i}|$

for $i=1,2,3,4$, where the terms $S_{i}$ are given by (3.4)-(3.7). To begin,
we define

$\alpha_{F,\pi}(\mathfrak{n})=\sum_{\begin{subarray}{c}\mathfrak{bc=n}\\\
\mathrm{N}\mathfrak{b}\leqslant X,~{}\mathrm{N}\mathfrak{c}\leqslant
X\end{subarray}}\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c}),\quad\beta_{F,\pi}(\mathfrak{n})=\sum_{\begin{subarray}{c}\mathfrak{bd=n}\\\
\mathfrak{b}>X\end{subarray}}\mu_{\pi}(\mathfrak{b})\lambda_{\pi}(\mathfrak{d}).$

By inequalities in Section 5, we can estimate the second moments of
$\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n}),\alpha_{F,\pi}(\mathfrak{n})$
and $\beta_{F,\pi}(\mathfrak{n})$. Firstly, we get from the Cauchy–Schwarz
inequality and (5.2) that

(9.1) $\displaystyle\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}|\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})|^{2}$
$\displaystyle\ll\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})\Big{)}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi\otimes\widetilde{\pi}}(\mathfrak{n})\Big{)}\ll
x.$

Secondly, by the bound (2.5) and the definitions of
$\Lambda_{F}(\mathfrak{n})$ and $a_{\pi\otimes\widetilde{\pi}}(\mathfrak{n})$,
we have

$\displaystyle\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}d_{F}(\mathfrak{n})\Lambda_{F}(\mathfrak{n})a_{\pi\otimes\widetilde{\pi}}(\mathfrak{n})$
$\displaystyle=\sum_{k\ll\log x}\sum_{\mathrm{N}\mathfrak{p}^{k}\leqslant
x}(k+1)a_{\pi\otimes\widetilde{\pi}}(\mathfrak{p}^{k})\log\mathrm{N}\mathfrak{p}$
$\displaystyle=\sum_{k\leqslant
n^{2}+1}\sum_{\mathrm{N}\mathfrak{p}^{k}\leqslant
x}(k+1)a_{\pi\otimes\widetilde{\pi}}(\mathfrak{p}^{k})\log\mathrm{N}\mathfrak{p}+x^{1-\frac{1}{n^{2}+1}+\varepsilon}$
$\displaystyle\ll\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi\otimes\widetilde{\pi}}(\mathfrak{n})+x^{1-\frac{1}{n^{2}+1}+\varepsilon}\ll
x.$

This further implies that

(9.2) $\displaystyle\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}|\alpha_{F,\pi}(\mathfrak{n})|^{2}$
$\displaystyle\ll\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\sum_{\mathfrak{bc=n}}|\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})|^{2}d_{F}(\mathfrak{n})$
$\displaystyle\ll(\log x)\sum_{\mathrm{N}\mathfrak{b}\leqslant
x}d_{F}(\mathfrak{b})|\mu_{\pi}(\mathfrak{b})|^{2}\sum_{\mathrm{N}\mathfrak{c}\leqslant
x/\mathrm{N}\mathfrak{b}}d_{F}(\mathfrak{c})\Lambda_{F}(\mathfrak{c})a_{\pi\otimes\widetilde{\pi}}(\mathfrak{c})\ll
x(\log x)^{n+2},$

by inserting Lemma 5.4, partial summation and the trivial fact
$d_{F}(\mathfrak{bc})\leqslant d_{F}(\mathfrak{b})d_{F}(\mathfrak{c})$.
Thirdly, we immediately obtain from Lemma 5.4 that

(9.3) $\displaystyle\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}|\beta_{F,\pi}(\mathfrak{n})|^{2}\ll x(\log x)^{4n+3}.$

###  9.1. The average for $S_{1}$

To estimate the contribution from $S_{1},$ we use the Cauchy–Schwarz
inequality and (9.1) to obtain

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{1}|\ll QX^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
X}|\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}\ll
QX.$

###  9.2. The average for $S_{2}$

We split $S_{2}$ in the following way:

$S_{2}=\Big{(}\sum_{\mathrm{N}\mathfrak{b}\leqslant
H}+\sum_{H<\mathrm{N}\mathfrak{b}\leqslant
X}\Big{)}\mu_{\pi}(\mathfrak{b})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{c}\leqslant
y/\mathrm{N}\mathfrak{b}\\\ \mathfrak{c}\equiv\mathfrak{ab^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c}=:S_{2}^{{}^{\prime}}+S_{2}^{{}^{\prime\prime}},$

where $H<X$. We deduce from partial summation and Theorem 1.3 that

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{2}^{{}^{\prime}}|$
$\displaystyle\leqslant\sum_{\mathrm{N}\mathfrak{b}\leqslant
H}|\mu_{\pi}(\mathfrak{b})|\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{ab},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{c}\leqslant
y/\mathrm{N}\mathfrak{b}\\\ \mathfrak{c}\equiv\mathfrak{ab^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c}\Big{|}$
$\displaystyle\ll\log x\Big{(}\sum_{\mathrm{N}\mathfrak{b}\leqslant
H}|\mu_{\pi}(\mathfrak{b})|\Big{)}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{c}\leqslant y\\\
\mathfrak{c}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{c})\Big{|}\ll\frac{Hx}{(\log
x)^{A}}.$

We use the Cauchy–Schwarz inequality and Lemma 5.4 in the last step, and
emphasize that $Q\leqslant x^{\frac{1}{\eta}}(\log x)^{-B}$ with
$\eta=\max\\{\frac{n}{2},2\\}$, $B=2^{\frac{n[F:\mathbb{Q}]}{4}}(2A+16)+2n-4$.
In order to estimate the contribution of $S_{2}^{{}^{\prime\prime}},$
splitting the range of summation over $\mathfrak{b}$ into intervals of the
form $\mathrm{N}\mathfrak{b}\sim M$ with $H<M\leqslant X$, we get

(9.4) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{2}^{{}^{\prime\prime}}|$ $\displaystyle\ll(\log x)\max_{H\leqslant
M\leqslant X}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}{\underset{\mathrm{N}\mathfrak{bc}\leqslant
y\atop\mathfrak{bc}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{b}\sim
M}\sum_{\mathrm{N}\mathfrak{c}\leqslant
x/M}}}\mu_{\pi}(\mathfrak{b})\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c}\Big{|}.$

In order to apply Lemma 8.2, we need to verify the sequence
$\\{\lambda_{\pi}(\mathfrak{n})\log\mathrm{N}\mathfrak{n}\\}$ satisfies the
Siegel–Walfisz hypothesis. So we have to estimate the sum of type
$\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})$ for any
$\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}$. We choose a function $h$
supported on $[0,x+Y]$, such that $h(z)=1$ if $Y\leqslant z\leqslant x$ and
$h^{(j)}(x)\ll_{j}Y^{-j}$ for all $j\geqslant 0$. Here, the parameter $Y$ will
be chosen later subject to $1\leqslant Y\leqslant x$. By partial integration,
the Mellin transform of $h$ satisfies

$\hat{h}(s)=\int_{0}^{x+Y}h(z)z^{s-1}dz\ll\frac{Y}{x^{1-\sigma}}\cdot\Big{(}\frac{x}{|s|Y}\Big{)}^{j}$

for any $j\geqslant 1$ and $1/2\leqslant\sigma=\mathop{\mathrm{Re}}s\leqslant
2$. Moreover, we derive from (2.10), Lemma 5.2 and the Cauchy–Schwarz
inequality that

$\sum_{x<\mathrm{N}\mathfrak{n}\leqslant x+Y}|\lambda_{\pi}(\mathfrak{n})|\ll
Y^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
x+Y}|\lambda_{\pi}(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}\ll(xY)^{\frac{1}{2}}$

for $0<Y\leqslant x$. Consequently, we have that

$\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})=\sum_{\mathfrak{n}}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})h(\mathrm{N}\mathfrak{n})+O((xY)^{\frac{1}{2}}).$

By Mellin’s inverse transform, we can write

(9.5)
$\sum_{\mathfrak{n}}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})h(\mathrm{N}\mathfrak{n})=\frac{1}{2\pi
i}\int_{(2)}\hat{h}(s)\Big{(}\sum_{\mathfrak{n}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}\Big{)}ds.$

We know from (7.4) that the Dirichlet series appearing above equals
$L(s,\pi\otimes\chi_{1})$ except for some Euler factors, where
$\chi_{1}(\text{mod}\;\mathfrak{m}^{\prime})$ is a primitive character induced
by $\chi$ with $\mathfrak{m}^{\prime}|\mathfrak{m}$. The convexity bound for
$L(s,\pi\otimes\chi_{1})$ gives

$L(\tfrac{1}{2}+it,\pi\otimes\chi_{1})\ll({\mathrm{N}\mathfrak{m}^{\prime}}(3+|t|)^{[F:\mathbb{Q}]})^{\frac{n}{4}+\varepsilon}.$

Moving the vertical line of integration in (9.5) to
$\mathop{\mathrm{Re}}s=1/2$, we obtain by Cauchy’s theorem and (7.4) that

$\sum_{\mathfrak{n}}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})h(\mathrm{N}\mathfrak{n})\ll
d_{F}(\mathfrak{m})^{2n}\int_{(1/2)}|\hat{h}(s)L(s,\pi\otimes\chi_{1})|ds\\\
\ll\int_{0}^{x/Y}\frac{x^{\frac{1}{2}}}{1+t}({\mathrm{N}\mathfrak{m}^{\prime}}(3+t)^{[F:\mathbb{Q}]})^{\frac{n}{4}}(\mathrm{N}\mathfrak{m}(3+t))^{\varepsilon}dt\\\
+\int_{x/Y}^{\infty}\frac{Y}{x^{\frac{1}{2}}}\cdot\Big{(}\frac{x}{tY}\Big{)}^{\frac{n[F:\mathbb{Q}]}{4}+2}({\mathrm{N}\mathfrak{m}^{\prime}}t^{[F:\mathbb{Q}]})^{\frac{n}{4}}(\mathrm{N}\mathfrak{m}t)^{\varepsilon}dt\ll\mathrm{N}\mathfrak{m}^{\varepsilon}{\mathrm{N}\mathfrak{m}^{\prime}}^{\frac{n}{4}}\Big{(}\frac{x}{Y}\Big{)}^{\frac{n[F:\mathbb{Q}]}{4}+\varepsilon}x^{\frac{1}{2}}.$

Gathering the above results we arrive at

$\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})\ll\mathrm{N}\mathfrak{m}^{\varepsilon}{\mathrm{N}\mathfrak{m}^{\prime}}^{\frac{n}{4}}\Big{(}\frac{x}{Y}\Big{)}^{\frac{n[F:\mathbb{Q}]}{4}+\varepsilon}x^{\frac{1}{2}}+(xY)^{\frac{1}{2}}.$

We choose
$Y=({\mathrm{N}\mathfrak{m}^{\prime}}x^{d})^{\frac{n}{n[F:\mathbb{Q}]+2}}$,
thus obtaining

$\sum_{n\leqslant
x}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})\ll\mathrm{N}\mathfrak{m}^{\frac{n}{2n[F:\mathbb{Q}]+4}+\varepsilon}x^{\frac{n[F:\mathbb{Q}]+1}{n[F:\mathbb{Q}]+2}+\varepsilon}.$

Since the orthogonality of characters yields

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant x\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\ll\max_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}\Big{|}\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})\Big{|}\leqslant
x^{1-\varepsilon}$

for any $\mathfrak{m}$ with $\mathrm{N}\mathfrak{m}\leqslant
x^{\frac{2}{n}-\varepsilon}$, the sequence $\\{\lambda_{\pi}(\mathfrak{n})\\}$
satisfies the Siegel–Walfisz hypothesis, and so does
$\\{\lambda_{\pi}(\mathfrak{n})\log\mathrm{N}\mathfrak{n}\\}$ by partial
summation. Thus, we can apply Lemma 8.2 to the sum on the last line of (9.4)
and then get from the second estimate in Lemma 5.4

$\displaystyle\ll(\log xQ)^{4}\max_{H\leqslant M\leqslant
X}\Big{(}Q+\sqrt{M+\frac{x}{M}}+\frac{\sqrt{x}}{(\log
x/M)^{A}}\Big{)}\sqrt{\sum_{\mathrm{N}\mathfrak{b}\sim
M}|\mu_{\pi}(\mathfrak{b})|^{2}\sum_{\mathrm{N}\mathfrak{c}\sim
x/M}|\lambda_{\pi}(\mathfrak{c})\log\mathfrak{c}|^{2}}$ $\displaystyle\ll(\log
xQ)^{5}\Big{(}Q\sqrt{x}+\sqrt{Xx}+\frac{x}{\sqrt{H}}+\frac{x}{(\log
x/X)^{A}}\Big{)}.$

To sum up, the contribution from $S_{2}$ is

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{2}|\ll\frac{Hx}{(\log x)^{A}}+(\log
xQ)^{5}\Big{(}Q\sqrt{x}+\sqrt{Xx}+\frac{x}{\sqrt{H}}+\frac{x}{(\log
x/X)^{A}}\Big{)}.$

###  9.3. The average for $S_{3}$.

The treatment is similar to that of $S_{2}$. If we use (9.2), then

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{3}|$ $\displaystyle\ll\frac{Hx}{(\log x)^{A-\frac{n}{2}-1}}+(\log
xQ)^{\frac{n}{2}+5}\Big{(}Q\sqrt{x}+X\sqrt{x}+\frac{x}{\sqrt{H}}+\frac{x}{(\log
x/X^{2})^{A}}\Big{)}.$

###  9.4. The average for $S_{4}$

This contribution is of the same form as $S_{2}^{{}^{\prime\prime}}$ except
the coefficients $\beta_{F,\pi}(\mathfrak{n})$ instead of
$\mu_{\pi}(\mathfrak{n})$ and $\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})$
instead of $\lambda_{\pi}(\mathfrak{n})\log\mathfrak{n}$. Since the
Siegel–Walfisz condition of $\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})$
follows from Corollary 4.8, the estimates (9.1) and (9.3) give that

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{4}|\ll(\log xQ)^{2n+6}\Big{(}Q\sqrt{x}+\frac{x}{\sqrt{X}}+\frac{x}{(\log
X)^{A}}\Big{)}.$

###  9.5. Finishing the proof

Finally, we collect all of our estimates for the averages of $S_{1}$, $S_{2}$,
$S_{3}$, and $S_{4}$, taking $H=(\log x)^{\frac{2}{3}(A+4)}$ and
$X=x^{\frac{1}{3}}$, to arrive at

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant x^{\frac{1}{\eta}}(\log
x)^{-B}}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})\Big{|}\ll\frac{x}{(\log
x)^{\frac{A}{3}-2n-6}},$

where
$\eta=\max\\{\frac{n}{2},2\\},B=2^{\frac{n[F:\mathbb{Q}]}{4}}(2A+16)+2n-4$.
After changing parameter $\frac{A}{3}-2n-6\mapsto A$ and inserting the
corresponding estimate into (3.3), Theorem 1.1 follows.

##  10\. An arithmetic application: Proof of Corollary 1.4

By the definition of divisor function, we have

$d(m)=2\sum_{\begin{subarray}{c}r\mid m\\\
r<\sqrt{m}\end{subarray}}1+\delta_{\square}(m),\qquad\delta_{\square}(m):=\left\\{\begin{array}[]{ll}1&\text{
if }m\text{ is a perfect square,}\\\ 0&\text{ otherwise.}\end{array}\right.$

Hence, we deduce that

(10.1) $\displaystyle\sum_{p\leqslant x}\lambda_{\pi}(p)d(p-1)$
$\displaystyle=2\sum_{p\leqslant
x}\lambda_{\pi}(p)\sum_{\begin{subarray}{c}r|(p-1)\\\
r<\sqrt{p-1}\end{subarray}}1+\sum_{p\leqslant
x}\lambda_{\pi}(p)\delta_{\square}(p-1)$
$\displaystyle=2\sum_{r<\sqrt{x-1}}\sum_{\begin{subarray}{c}p\leqslant x\\\
p\equiv
1\,(\operatorname{mod}{r})\end{subarray}}\lambda_{\pi}(p)+O(x^{\frac{3}{4}}).$

Here the error term comes from

$\sum_{p\leqslant
x}\lambda_{\pi}(p)\delta_{\square}(p-1)\ll\Big{(}\sum_{p\leqslant
x}|\lambda_{\pi}(p)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{p\leqslant
x}\delta_{\square}(p-1)\Big{)}^{\frac{1}{2}}\ll x^{\frac{3}{4}}$

by using the Cauchy–Schwarz inequality and the Rankin–Selberg theory.

For $r\leqslant x^{\frac{1}{2}}(\log x)^{-B}$, the result in Theorem 1.1
produces a small enough estimate. Then we have

(10.2) $\displaystyle\sum_{r\leqslant x^{\frac{1}{2}}(\log
x)^{-B}}\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{r})\end{subarray}}\lambda_{\pi}(p)\ll\sum_{r\leqslant
x^{\frac{1}{2}}(\log x)^{-B}}\Big{|}\sum_{\begin{subarray}{c}p\leqslant x\\\
p\equiv
1\,(\operatorname{mod}{r})\end{subarray}}\lambda_{\pi}(p)\Big{|}\ll\frac{x}{(\log
x)^{A}}.$

For the contribution of the terms $\sqrt{x}(\log x)^{-B}<r<\sqrt{x-1},$ the
Brun-Titchmarsh inequality is used in general. However, it is not suitable for
the automorphic context, since GRC remains open. By using the orthogonality
relation of additive characters, we get

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{r})\end{subarray}}\lambda_{\pi}(p)=\frac{1}{r}\sum_{q|r}\sideset{}{{}^{*}}{\sum}_{a\,(\operatorname{mod}{q})}e\Big{(}\frac{-a}{q}\Big{)}\sum_{p\leqslant
x}\lambda_{\pi}(p)e\Big{(}\frac{ap}{q}\Big{)}.$

Thus,

(10.3) $\displaystyle\sum_{\frac{\sqrt{x}}{(\log
x)^{B}}<r<\sqrt{x-1}}\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{r})\end{subarray}}\lambda_{\pi}(p)$
$\displaystyle=\sum_{l<\sqrt{x-1}}\frac{1}{l}\sum_{\frac{\sqrt{x}}{l(\log
x)^{B}}<q<\frac{\sqrt{x-1}}{l}}\frac{1}{q}\;\;\sideset{}{{}^{*}}{\sum}_{a\,(\operatorname{mod}{q})}e\Big{(}\frac{-a}{q}\Big{)}\sum_{p\leqslant
x}\lambda_{\pi}(p)e\Big{(}\frac{ap}{q}\Big{)}:=T_{1}+T_{2},$

where $T_{1}$ denotes the contribution of the terms $l\leqslant(\log x)^{B}$
and $T_{2}$ denotes the contribution of the other terms. First we treat the
sum $T_{1}$. Applying the Cauchy–Schwarz inequality and the additive large
sieve inequality (e.g. [16, Theorem 7.11]), we have

(10.4) $\displaystyle T_{1}$ $\displaystyle\ll\sum_{l\leqslant(\log
x)^{B}}\frac{1}{l}\sum_{\frac{\sqrt{x}}{l(\log
x)^{B}}<q<\frac{\sqrt{x+1}}{l}}\frac{1}{q}\;\;\sideset{}{{}^{*}}{\sum}_{a\,(\operatorname{mod}{q})}\Big{|}\sum_{p\leqslant
x}\lambda_{\pi}(p)e\Big{(}\frac{ap}{q}\Big{)}\Big{|}$
$\displaystyle\ll\sum_{l\leqslant(\log
x)^{B}}\frac{1}{l}\Big{(}\sum_{q<\frac{\sqrt{x+1}}{l}}\;\;\sideset{}{{}^{*}}{\sum}_{a\,(\operatorname{mod}{q})}\Big{|}\sum_{p\leqslant
x}\lambda_{\pi}(p)e\Big{(}\frac{ap}{q}\Big{)}\Big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\frac{\sqrt{x}}{l(\log
x)^{B}}<q<\frac{\sqrt{x+1}}{l}}\frac{1}{q}\Big{)}^{\frac{1}{2}}$
$\displaystyle\ll x^{\frac{1}{2}}\Big{(}\sum_{p\leqslant
x}|\lambda_{\pi}(p)|^{2}\Big{)}^{\frac{1}{2}}(\log\log
x)^{\frac{3}{2}}\ll\frac{x(\log\log x)^{\frac{3}{2}}}{\sqrt{\log x}},$

where we use the fact that $|\lambda_{\pi}(p)|^{2}\ll
a_{\pi\times\widetilde{\pi}}(p)$ and the estimate (5.2) in the last step.
Next, we treat the sum $T_{2}$. Interchanging the order of summation and
applying the formula for the Ramanujan sum

$\sideset{}{{}^{*}}{\sum}_{a\,(\operatorname{mod}{q})}e\Big{(}\frac{(p-1)a}{q}\Big{)}=\sum_{d|(p-1,q)}d\mu\Big{(}\frac{q}{d}\Big{)},$

we obtain from Theorem 1.1 that

(10.5) $T_{2}=\sum_{(\log
x)^{B}<l<\sqrt{x-1}}\frac{1}{l}\sum_{m<\frac{\sqrt{x-1}}{l}}\frac{\mu(m)}{m}\sum_{\frac{\sqrt{x}}{ml(\log
x)^{B}}<q<\frac{\sqrt{x-1}}{ml}}\sum_{\begin{subarray}{c}p\leqslant x\\\
p\equiv 1\,(\operatorname{mod}{q})\end{subarray}}\lambda_{\pi}(p)\\\ \ll(\log
x)^{2}\sum_{q<\frac{\sqrt{x-1}}{(\log
x)^{B}}}\Big{|}\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{q})\end{subarray}}\lambda_{\pi}(p)\Big{|}\ll\frac{x}{(\log
x)^{A}}.$

Assembling these estimates in (10.2)–(10.5) yields

$\sum_{r<\sqrt{x-1}}\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{r})\end{subarray}}\lambda_{\pi}(p)\ll\frac{x(\log\log
x)^{\frac{3}{2}}}{\sqrt{\log x}}.$

Inserting this into (10.1), this corollary follows.

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######  $G_{6}$ ¡–¿ $G_{7}$.

If $\mathcal{A}$ can distinguish between $G_{6}$ and $G_{7}$, then we can
build a distinguisher $\mathcal{B}$ that breaks the ind$-cpa property of
$(\textsf{Enc},\textsf{Dec})$. In the ind$-cpa game [49], $\mathcal{B}$
receives:

  * •

a challenge ciphertext $c_{\textsf{payload}}=c_{b}$, s.t.
$c_{0}=\textsf{Enc}_{K_{enc}}(m)$ on a chosen-by-$\mathcal{B}$ $m$,
$c_{1}\overset{\$}{\leftarrow}\\{0,1\\}^{|c_{0}|}$, and
$b\overset{\$}{\leftarrow}\\{0,1\\}$.

$\mathcal{B}$ runs MsPURB.Dec as before to create a challenge for
$\mathcal{A}$, except that $\mathcal{B}$ uses the ind$-cpa challenge
ciphertext $c_{\textsf{payload}}$ in step (7), instead of encrypting, as
$\mathcal{B}$ does not know $K_{enc}$. As before, $\mathcal{A}$’s view is
independent of $K^{\star}$, so also this change of $K_{enc}$ remains
undetected.

$\mathcal{B}$ answers decryption queries $\textsf{qDec}(pk_{i}(S_{i}),c)$ from
$\mathcal{A}$ as before. In particular

  * •

if $\texttt{Unhide}(\tau)=X_{i}^{\star}$ and $e_{i}=e^{\star}_{i}$,
$\mathcal{B}$ returns $\bot$ as per the changes in $G_{5}$;

  * •

Otherwise, $\mathcal{B}$ runs $\textsf{MsPURB.Dec}(\cdot)$.

If $b=0$, $\mathcal{B}$ simulates $G_{6}$, and, if $b=1$, $\mathcal{B}$
simulates $G_{7}$. Hence, if $\mathcal{A}$ can distinguish between these two
games, $\mathcal{B}$ can break the the ind$-cpa property of
$(\textsf{Enc},\textsf{Dec})$. More precisely:

$\big{|}\textrm{Pr}[W_{6}]-\textrm{Pr}[W_{7}]\big{|}\leq\textsf{Adv}^{\textsf{ind\$-cpa}}_{(\textsf{Enc},\textsf{Dec}),\mathcal{A}}(1^{\lambda}).$

Combining the individual inequalities we find that there exists adversaries
$\mathcal{B}_{1},\ldots,\mathcal{B}_{5}$ such that

$\displaystyle\textsf{Adv}^{\textsf{cca2-out}}_{\textsf{msbe},\mathcal{A}}(1^{\lambda})\leq\;$
$\displaystyle
r\left(\textsf{Adv}^{\textsf{cca2}}_{\textsf{KEM},\mathcal{B}_{1}}(1^{\lambda})+\textsf{Adv}^{\textsf{ind\$-cca2}}_{\Pi,\mathcal{B}_{2}}(1^{\lambda})\right)+$
$\displaystyle\textsf{Adv}^{\textsf{suf}}_{\textsf{MAC},\mathcal{B}_{3}}(1^{\lambda})+\textsf{Adv}^{\textsf{ind\$}}_{\textsf{MAC},\mathcal{B}_{4}}(1^{\lambda})+$
$\displaystyle\textsf{Adv}^{\textsf{ind\$-cpa}}_{(\textsf{Enc},\textsf{Dec}),\mathcal{B}_{5}}(1^{\lambda}),$

completing the proof. ∎

###  D.3 Proof of Theorem 2

For our MsPURB ind$-cpa recipient-privacy game, we take inspiration from the
single-suite recipient-privacy game defined by Barth et al. [4], but we
restate it in the ind$-cpa setting.

######  Game Recipient-Privacy.

The game is between a challenger and an adversary $\mathcal{A}$, and proceeds
along the following phases:

Init: The challenger and adversary take $\lambda$ as input. The adversary
outputs a number of recipients $r$ and corresponding cipher suites
$S_{1},\ldots,S_{r}$ it wants to attack. Let $s$ be the number of unique
cipher suites. The challenger verifies, for each $i\in\\{1,\ldots,r\\}$, that
$S_{i}$ is a valid cipher suite, i.e., that it a valid output of
$\textsf{MSBE.}\textsf{Setup}(1^{\lambda})$. The challenger aborts, and sets
$b^{\star}\overset{\$}{\leftarrow}\\{0,1\\}$ if the suites are not all valid.
Adversary $\mathcal{A}$ then outputs two sets of recipients
$N_{0},N_{1}\subseteq\\{1,\ldots,n\\}$ such that $|N_{0}|=|N_{1}|=r$, and the
number of users in $N_{0}$ and $N_{1}$ using suite $S_{j}$ is the same.

Setup: For each $i\in 1,\ldots,n$ given by $\mathcal{A}$, the challenger runs
$(sk_{i},pk_{i})\leftarrow\textsf{MsPURB.KeyGen}(S_{i})$, where $S_{i}$ is
previously chosen by $\mathcal{A}$. The challenger gives two sets
$R_{0}=\\{pk^{0}_{1},\ldots,pk^{0}_{r}\\}$ and
$R_{1}=\\{pk^{1}_{1},\ldots,pk^{1}_{r}\\}$ to $\mathcal{A}$, where
$R_{0},R_{1}$ are the generated public keys of the recipients $N_{0},N_{1}$
respectively. The challenger also gives to $\mathcal{A}$ all $sk_{i}$ that
correspond to $i\in N_{0}\cap N_{1}$.

Challenge: $\mathcal{A}$ outputs $m^{\star}$. The challenger generates
$c_{0}=\textsf{MsPURB.Enc}(R_{0},m^{\star})$ and
$c_{1}=\textsf{MsPURB.Enc}(R_{1},m^{\star})$. The challenger flips a coin
$b\overset{\$}{\leftarrow}\\{0,1\\}$ and sends $c^{\star}=c_{b}$ to
$\mathcal{A}$.

Guess: $\mathcal{A}$ outputs its guess $b^{\star}$ for $b$ and wins if
$b^{\star}=b$.

We define $\mathcal{A}$’s advantage in this game as:

$\textsf{Adv}^{\textsf{cpa-
in}}_{\textsf{msbe},\mathcal{A}}(1^{\lambda})=2\left|\textrm{Pr}[b=b^{\star}]-\tfrac{1}{2}\right|.$

We say that a MSBE scheme is cpa-secure against insiders if
$\textsf{Adv}^{\textsf{cpa-in}}_{\textsf{msbe},\mathcal{A}}(1^{\lambda})$ is
negligible in the security parameter.

The conditions on $N_{0}$ and $N_{1}$ in the game ensure that $\mathcal{A}$
cannot trivially win by looking at the size of the ciphertext. PURBs allows
for suites with different groups (resulting in different size encodings of the
corresponding IES public key) and for suites to use different authenticated
encryption schemes (that could result in different sizes of encrypted entry
points). Since PURBs must encode groups and entry points into the header, we
mandate that for each suite the number of recipients is the same in $N_{0}$
and $N_{1}$. This assumption is similar to requiring equal-size sets of
recipients in a challenge game for single-suite broadcast encryption [4]. As
in broadcast encryption, if this requirement is an issue, a sender can add
dummy recipients to avoid structural leakage to an insider adversary.

We will show that

$\textsf{Adv}^{\textsf{cpa-in}}_{\textsf{msbe},\mathcal{A}}(1^{\lambda})\leq
2d\cdot\textsf{Adv}^{\textsf{cca2}}_{\textsf{KEM},\mathcal{B}}(1^{\lambda}),$

where $d$ is the number of recipients in which $N_{0}$ and $N_{1}$ differ.

###### Proof.

Similarly to Barth et al. [4], we prove recipient privacy when the sets
$R_{0}$ and $R_{1}$ differ only by one public key in one suite. The general
case follows by a hybrid argument. Consider the following games:

######  Game $G_{0}$.

This game is as the original recipient-privacy ind$-cpa game where $b=0$ and
$pk_{i}=R_{0}\setminus R_{1}$, $pk_{j}=R_{1}\setminus R_{0}$, where the public
keys $pk_{i}$ and $pk_{j}$ are of the same suite $S$.

######  Game $G_{1}$.

As in $G_{0}$, but we change how a key $k_{i}^{\star}$ corresponding to the
recipient $i$ is computed in HdrPURB.Encap for the _challenge ciphertext_.
Instead of computing $k_{i}^{\star}=\textnormal{H}(Y_{i}^{x})$ (where
$Y_{i}=pk_{i}$) as in step (2) of HdrPURB.Encap, we set
$k_{i}^{\star}\overset{\$}{\leftarrow}\\{0,1\\}^{\lambda_{H}}$. As the
challenger generates fresh public keys for each encryption query and thus a
fresh key $k_{i}$, and does not have to answer decryption queries, it does not
need to memorize $k_{i}^{\star}$.

######  Game $G_{2}$.

As in $G_{1}$, but we change the random sampling $k_{i}^{\star}$ in
HdrPURB.Encap for the challenge ciphertext with
$k_{i}^{\star}=\textnormal{H}(Y_{j}^{x})=k_{j}^{\star}$ where $Y_{j}=pk_{j}$.
The game now is the original recipient-privacy ind$-cpa game where $b=1$.

Conclusion. $G_{0}$ represents the recipient-privacy game with $b=0$ and
$G_{2}$ recipient-privacy game with $b=1$. If $\mathcal{A}$ cannot distinguish
between $G_{0}$ and $G_{2}$, $\mathcal{A}$ does not have an advantage in
winning the recipient-privacy game.

Let $W_{i}$ be the event that $\mathcal{A}$ outputs $b^{\star}=1$ in game
$G_{i}$.

######  $G_{0}$ ¡–¿ $G_{1}$.

If $\mathcal{A}$ can distinguish between $G_{0}$ and $G_{1}$, we can build a
distinguisher $\mathcal{B}$ against the ind$-cca2 security of the IES KEM.
Recall that $\mathcal{B}$ receives, from its ind$-cca2-KEM challenger,

  * •

a public key $Y$;

  * •

a challenge $\langle X^{\star},k^{\star}\rangle$, where depending on bit
$b\overset{\$}{\leftarrow}\\{0,1\\}$, we have
$k^{\star}=\textnormal{H}({Y}^{x^{\star}})$ if $b=0$ or
$k^{\star}\overset{\$}{\leftarrow}\\{0,1\\}^{l(\lambda)}$ if $b=1$;

  * •

access to a $\textsf{Decap}(\cdot)$ oracle for all but $X^{\star}$.

At the start of the game, $\mathcal{B}$ will set $pk_{i}=Y$, so that the
public key of recipient $i$ matches that of its IES KEM challenger. Note that
$\mathcal{B}$ does not know the corresponding private key $y_{i}$. For all
other recipients $h$, $\mathcal{B}$ sets
$(sk_{h}=y_{h},pk_{h}=Y_{h})=\textsf{MsPURB.KeyGen}(S_{h})$. As $\mathcal{A}$
plays an ind$-cpa game, $\mathcal{B}$ does not need to use the
$\textsf{Decap}(\cdot)$ oracle (in fact, for ind$-cpa recipient privacy
ind$-cpa security of the IES KEM suffices).

If $b=0$ in the IES-KEM challenge, then $\mathcal{B}$ simulates $G_{0}$, and,
If $b=1$, $\mathcal{B}$ simulates $G_{1}$. Hence, if $\mathcal{A}$
distinguishes between $G_{0}$ and $G_{1}$, $\mathcal{B}$ wins in the ind$-cca2
IES-KEM game. Therefore:

$\left|\textrm{Pr}[W_{0}]-\textrm{Pr}[W_{1}]\right|\leq\textsf{Adv}^{\textsf{cca2}}_{\textsf{KEM},\mathcal{B}}(1^{\lambda})$

######  $G_{1}$ ¡–¿ $G_{2}$.

The proof follows the same steps as the proof of $G_{0}$ ¡–¿ $G_{1}$.
Therefore:

$\left|\textrm{Pr}[W_{0}]-\textrm{Pr}[W_{1}]\right|\leq\textsf{Adv}^{\textsf{cca2}}_{\textsf{KEM},\mathcal{B}}(1^{\lambda}).$

Let $d$ be the number of recipients that differ in $N_{0}$ and $N_{1}$. Then
by repeating the above two steps $d$ times in a hybrid argument, we find that:

$\textsf{Adv}^{\textsf{cpa-in}}_{\textsf{msbe},\mathcal{A}}(1^{\lambda})\leq
2d\cdot\textsf{Adv}^{\textsf{cca2}}_{\textsf{KEM},\mathcal{B}}(1^{\lambda}),$

as desired. ∎