  Motivated by recent interest in federated submodel learning, this work
explores the fundamental problem of privately reading from and writing to a
database comprised of $K$ files (submodels) that are stored across $N$
distributed servers according to an $X$-secure threshold secret sharing scheme.
One after another, various users wish to retrieve their desired file, locally
process the information and then update the file in the distributed database
while keeping the identity of their desired file private from any set of up to
$T$ colluding servers. The availability of servers changes over time, so
elastic dropout resilience is required. The main contribution of this work is
an adaptive scheme, called ACSA-RW, that takes advantage of all currently
available servers to reduce its communication costs, fully updates the database
after each write operation even though the database is only partially
accessible due to server dropouts, and ensures a memoryless operation of the
network in the sense that the storage structure is preserved and future users
may remain oblivious of the past history of server dropouts. The ACSA-RW
construction builds upon CSA codes that were originally introduced for XSTPIR
and have been shown to be natural solutions for secure distributed matrix
multiplication problems. ACSA-RW achieves the desired private read and write
functionality with elastic dropout resilience, matches the best results for
private-read from PIR literature, improves significantly upon available
baselines for private-write, reveals a striking symmetry between upload and
download costs, and exploits redundant storage dimensions to accommodate
arbitrary read and write dropout servers up to certain threshold values. It
also answers in the affirmative an open question by Kairouz et al. by
exploiting synergistic gains from the joint design of private read and write
operations.
