What adsl router to buy a budget to $100?

Which adsl (ADSL2+) router to buy a budget to $100 (c wi-fi, several LAN ports, support for hardware AES, hardware encryption for VPN tunnels)?

One step at a time...

I generally suggest that a separate modem is a good idea, as that could be a limiting factor on your bandwidth and latency. Also, as technologies change, or you change your ISP, you aren't "throwing away" your router as well.

Second, those are some pretty steep requirements for even a router in the $100 price range, without an integrated modem. You're asking for high-end encryption features and performance that are pushing you into the "enthusiast" class of devices, if not into mid-range amd64 / x86_64 hardware. Sub-$100 devices probably can't push more than 20-50 Mbps over VPN encryption, even with the faster ciphers today.

Which is more important to you, budget or encryption performance?

There is probably just one really plattform that we can recommend today: The Lantiq XRX200. It supports ADSL2+ (Annex A, Annex B, Annex J, Annex M, ..) and supports VDSL, VDSL2, VDSL2 Vectoring. It also supports the 30a VDSL2 Profile.
What is the country you are from? In many parts of europe you can get used ISP devices from ebay for about 1$.
If you need good wifi (only ath9k is really free and good wifi), then take a look into the TP-Link TD-W8970. On ADSL Lines you have to care about the Annex format. Some have soldered resistor for Annex A, Some have soldered resistor for Annex B. You have to solder if you get the wrong hardware version for your country.

As long as only ADSL is required, the WAN speed (and with that the maximum expectation for VPN tunnels) is limited to 20 MBit/s (and <16 MBit/s in practice), this is easy to accomplish with lantiq hardware - without hardware AES.

For ADSL Annex A the best option would be the BT Home Hub 5 Type A (read the wiki before buying, the initial flashing is a bit involved and it might make sense to buy a device with LEDE preflashed), you'll find them for <10 GBP with the vendor firmware and ~25+ GBP for preflashed devices.

For ADSL Annex B the AVM Fritz!Box 3370 (rev 2!) might be a pretty good option for ~25 EUR.

Both devices will support VDSL2 up to 100 MBit/s (with flow-offloading enabled, hard at the limit, but possible), VPN won't work at that performance levels (at all), but sufficiently for <= 16 MBit/s (as in ADSL).

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if I did not have adsl, then I could buy this router

The MikroTik hAP ac² is defenetly not a device i would recommend. It requires closed source software to run the wifi, it have "just" 16MB of flash and many other negative things.

The best free device is the Netgear WNDR4700. It have 3TX3RX ath9k 2,4Ghz and 3TX3RX ath9k 5ghz. This combined with SATA port, fast cpu, enough RAM and ROM, USB 3.0, and many other great things.
But thats not the topic of this Forum post here. Its about the ADSL router.

@slh have made a good post. I would simply recommend the TP-Link TD-W8970 if 2,4ghz (ath9k) is enough. The TP-Link TD-W8970 is available in Annex A and Annex B and just works.

I'm from Russia.
judging by the outgoing speed of annex A

Then you can get for example this one from USA for total under 100$ : https://www.ebay.com/itm/223092921729

The TP-Link TD-W8970 is simple to differ between Annex A and Annex B.
Annex A: TP-Link TD-W8970
Annex B: TP-Link TD-W8970B

Watch out to not get some broadcom crap that is inside the hardware revision 3 (the one with the 2 antennas). The one i linked above have Revision 1.2 .Thats fine and i used it many times.

For complete explaination in this forum post:
You dont have to care about the Annex version on xrx200 devices if you use the device on a VDSL line. Thats why i can use Annex A and B here when using a VDSL line.

The point being that ADSL Annexes are not the same as VDSL Anexxes, For ADSL Annex A ist for most of the world, while Annex B is for coexistence with ISDN (mainly Germany?), it leaves out a few more low frequencies to allow ISDN to continue to work; for VDSL all Bandplans (so which frequencies are used) for Europe are described in Annex B, for North Amerika in Annex A, and for Japan? in Annex C. I assume since the actual bandplans all are pretty similar (or rather have large overlaps) there should be no hardware difference for VDSL devices (unlike ADSL that ofen use different fixed high-pass filters that might require a bit of soldering to change from Annex A to Annex B and vice versa).

Thanks for the detailed answer and recommendation. BT Home Hub 5A, then what you need.