Please post the exact command you're running and the exact output

`sudo systemctl start fprintd.service`

```
systemd[1]: Starting Fingerprint Authentication Daemon...
fprintd[1296]: /usr/lib/fprintd: error while loading shared libraries: libcrypto.so.1.1: cannot open shared object file: No such file or direct>
systemd[1]: fprintd.service: Main process exited, code=exited, status=127/n/a
systemd[1]: fprintd.service: Failed with result 'exit-code'.
systemd[1]: Failed to start Fingerprint Authentication Daemon.
```

Installing the package `openssl-1.1` fixes it for me.

What is the output of lddtree /usr/lib/fprintd (requires pax-utils)
This is the output on my system, it does not show any linkage to libcrypto
/usr/lib/fprintd (interpreter => /lib64/ld-linux-x86-64.so.2)
libglib-2.0.so.0 => /usr/lib/libglib-2.0.so.0
libpcre2-8.so.0 => /usr/lib/libpcre2-8.so.0
libgio-2.0.so.0 => /usr/lib/libgio-2.0.so.0
libz.so.1 => /usr/lib/libz.so.1
libmount.so.1 => /usr/lib/libmount.so.1
libblkid.so.1 => /usr/lib/libblkid.so.1
libgobject-2.0.so.0 => /usr/lib/libgobject-2.0.so.0
libffi.so.8 => /usr/lib/libffi.so.8
libgmodule-2.0.so.0 => /usr/lib/libgmodule-2.0.so.0
libfprint-2.so.2 => /usr/lib/libfprint-2.so.2
libgusb.so.2 => /usr/lib/libgusb.so.2
libusb-1.0.so.0 => /usr/lib/libusb-1.0.so.0
libudev.so.1 => /usr/lib/libudev.so.1
libatomic.so.1 => /usr/lib/libatomic.so.1
libjson-glib-1.0.so.0 => /usr/lib/libjson-glib-1.0.so.0
libm.so.6 => /usr/lib/libm.so.6
libpixman-1.so.0 => /usr/lib/libpixman-1.so.0
libnss3.so => /usr/lib/libnss3.so
libnssutil3.so => /usr/lib/libnssutil3.so
libplds4.so => /usr/lib/libplds4.so
libplc4.so => /usr/lib/libplc4.so
libnspr4.so => /usr/lib/libnspr4.so
libgudev-1.0.so.0 => /usr/lib/libgudev-1.0.so.0
libpolkit-gobject-1.so.0 => /usr/lib/libpolkit-gobject-1.so.0
libsystemd.so.0 => /usr/lib/libsystemd.so.0
libcap.so.2 => /usr/lib/libcap.so.2
libgcrypt.so.20 => /usr/lib/libgcrypt.so.20
libgpg-error.so.0 => /usr/lib/libgpg-error.so.0
liblzma.so.5 => /usr/lib/liblzma.so.5
libzstd.so.1 => /usr/lib/libzstd.so.1
liblz4.so.1 => /usr/lib/liblz4.so.1
libgcc_s.so.1 => /usr/lib/libgcc_s.so.1
libc.so.6 => /usr/lib/libc.so.6

Thank you for the hint.
So it seems that it is needed by `libfprint-2.so.2`

```
$ pacman -F libfprint-2.so.2
extra/libfprint 1.94.5-1 (fprint)
usr/lib/libfprint-2.so.2
```

```
$ lddtree /usr/lib/fprintd
/usr/lib/fprintd (interpreter => /lib64/ld-linux-x86-64.so.2)
libc.so.6 => /usr/lib/libc.so.6
libdbus-glib-1.so.2 => /usr/lib/libdbus-glib-1.so.2
libdbus-1.so.3 => /usr/lib/libdbus-1.so.3
libsystemd.so.0 => /usr/lib/libsystemd.so.0
libcap.so.2 => /usr/lib/libcap.so.2
libgcrypt.so.20 => /usr/lib/libgcrypt.so.20
libgpg-error.so.0 => /usr/lib/libgpg-error.so.0
liblzma.so.5 => /usr/lib/liblzma.so.5
libzstd.so.1 => /usr/lib/libzstd.so.1
liblz4.so.1 => /usr/lib/liblz4.so.1
libgcc_s.so.1 => /usr/lib/libgcc_s.so.1
libgobject-2.0.so.0 => /usr/lib/libgobject-2.0.so.0
libffi.so.8 => /usr/lib/libffi.so.8
libglib-2.0.so.0 => /usr/lib/libglib-2.0.so.0
libpcre2-8.so.0 => /usr/lib/libpcre2-8.so.0
libgio-2.0.so.0 => /usr/lib/libgio-2.0.so.0
libz.so.1 => /usr/lib/libz.so.1
libmount.so.1 => /usr/lib/libmount.so.1
libblkid.so.1 => /usr/lib/libblkid.so.1
libgmodule-2.0.so.0 => /usr/lib/libgmodule-2.0.so.0
libfprint-2.so.2 => /usr/lib/libfprint-2.so.2
libgusb.so.2 => /usr/lib/libgusb.so.2
libusb-1.0.so.0 => /usr/lib/libusb-1.0.so.0
libudev.so.1 => /usr/lib/libudev.so.1
libatomic.so.1 => /usr/lib/libatomic.so.1
libjson-glib-1.0.so.0 => /usr/lib/libjson-glib-1.0.so.0
libpixman-1.so.0 => /usr/lib/libpixman-1.so.0
libm.so.6 => /usr/lib/libm.so.6
libnss3.so => /usr/lib/libnss3.so
libnssutil3.so => /usr/lib/libnssutil3.so
libplds4.so => /usr/lib/libplds4.so
libplc4.so => /usr/lib/libplc4.so
libnspr4.so => /usr/lib/libnspr4.so
libcrypto.so.1.1 => /usr/lib/libcrypto.so.1.1
libpolkit-gobject-1.so.0 => /usr/lib/libpolkit-gobject-1.so.0
```

Ah, I think I found the reason.
https://aur.archlinux.org/packages/libfprint-vfs009x-git needs it.
I'll close this issue ticket.

Thank you!