Gemara

One may not send a thigh to a Nochri (without removing the Gid), lest another Yisrael see and buy it from the Nochri (assuming that it is permitted). Alternatively, it is Geneivas Da'as (deception. The Nochri thinks that Reuven gave him something Reuven himself would eat.)

(Shmuel): It is forbidden to deceive people, even Nochrim.

Bava Kama 113a (Mishnah): We may falsely vow to tax collectors.

Question: Shmuel taught that the law of the kingdom is binding!

Answer #1 (Rav Chanina bar Kahana): The Mishnah discusses a tax collector who takes as much as he wants.

Answer #2 (Tana d'Vei R. Yishmael): It discusses a self-appointed tax collector.

Answer #3 (Rav Ashi): It discusses a Nochri tax collector.

Beraisa - R. Yishmael): If a Yisrael and a Nochri [extortionist] came for judgment, If Torah law favors the Yisrael, we rule like Torah law. If Nochri law favors the Yisrael, we rule like Nochri law. If neither favors the Yisrael, we seek a ruse;

R. Akiva forbids. A ruse is not Kidush Hash-m!

Question: If not for Kidush Hash-m, R. Akiva would permit a ruse. One may not steal from Nochrim!

(Beraisa - R. Shimon citing R. Akiva): We learn that Gezel Nochri is forbidden from "he (a Yisrael slave sold to a Nochri) will have redemption" - we may not forcibly take him without paying.

Suggestion: Perhaps he may trick him!

Rejection: "V'Chishev Im Konehu" - he will calculate with the buyer, i.e. precisely.

Answer (Rava): One may not steal from a Nochri, but one may use a ruse to evade paying a debt to a Nochri.

Question (Abaye): One must pay the proper redemption to a Nochri, even though this is like a debt!

Answer: Rava holds that a Yisrael slave is bodily owned by his master (for the period of service, so to forcibly take him is outright theft).

(Shmuel): If the Nochri erred, it is permitted (to keep it).

A Kusi (thought that he) sold a copper bucket to Shmuel for four Zuz. Really, it was gold. Shmuel covered up a Zuz. The Kusi did not notice.

Rav Kahana bought 120 Kelim from a Kusi for 100. He paid only 99, and said him 'I rely on you.' (Some explain that the Kusi said this to Rav Kahana.)

Ravina bought a date tree in partnership with a Kusi. Each would take half the wood. He told his servant to take from the trunk (before the Kusi takes), because the Kusi only looks at the number of logs, not the thickness.

Rishonim

Rif and Rosh (10:12): The Halachah follows Rav Ashi.

Rosh: One may uproot a Nochri's loan. Evading the tax is like uprooting his loan.

Rambam (Hilchos Gezeilah 1:2): Mid'Oraisa, one may not steal any amount, or be Oshek even a Nochri idolator. If one did, he must return it.

Question (Kesef Mishneh): It says "Lo Sa'asok Re'echa", to exclude a Nochri! Also, the Rambam explains that Oshek is uprooting a loan or wages that he owes. The Gemara permits this!

Answer (Kesef Mishneh): The Rambam holds like Rashi, that uprooting a loan that is not total theft is permitted when there is no Chilul Hash-m, e.g. he tells the Nochri's heir 'I paid your father.'

Rambam (11:4): A mistake of a Nochri is permitted, if he himself erred. One may not trick him.

Rambam (5): It is permitted if a Nochri erred in the calculation. The Yisrael must say 'I rely on your calculation. I don't know; I give what you tell me.' If he does not say this it is forbidden, lest the Nochri intends to check him, and it will be Chilul Hash-m.

Hagahos Maimoniyos (2): R. Chananel and Avi ha'Ezri rule similarly. One may not even deceive a Nochri (Chulin 94a)!

Rambam (Hilchos Geneivah 7:8): One may not deceive a Nochri about a calculation - "he will calculate (precisely) with the master", even if the Nochri is under your rule.

Yam Shel Shlomo (Bava Kama 10:20): Rashi explains that Shmuel bought a gold bucket in lieu of a copper bucket, and also withheld one Zuz from the price. Also Tosfos and the Tur permit tricking a Nochri if he will not notice. I disagree. We permit only when he himself erred! One may uproot a loan, i.e. a tax or debt. To pay less than the price is like theft, like taking by force. Also, it is forbidden due to Chilul Hash-m. Perhaps the Nochri intends to check, like the Rambam says! Rather, the Aruch and Ran explain that Shmuel gave an extra coin to the Nochri, so he would rejoice and go quickly (and not realize that he sold a gold bucket). It was permitted, for the Nochri erred by himself. Rashi must explain that since he acquired the bucket through Meshichah, the price was like a debt, which one may uproot. He agrees that one may not trick the Nochri into giving more Kelim or money than he should, even without Chilul Hash-m. In Hilchos Geneivah, the Rambam forbids tricking a Nochri about a calculation. He learns from "v'Chishev Im Konehu", and does not mention Chilul Hash-m. In Hilchos Gezeilah, he forbids lest the Nochri check and there will be Chilul Hash-m. He does not bring the verse. This is like I explained. We learn from the verse Gezel Nochri, for a master bodily acquires an Eved Ivri. The Gemara is difficult for Tosfos. We inferred that R. Akiva would permit a ruse if not for Kidush Hash-m, and asked from Eved Ivri. According to Tosfos, there is concern for Chilul Hash-m also regarding Eved Ivri (the Nochri is aware)! Rather, the verse forbids deceiving a Nochri even if he will not know. Ravina told his servant to take the thick branches, i.e. before the Kusi made a proper Kinyan. After this, one may not take a bigger share, for it is Gezel, and one may not trick him.

Poskim

Rema (CM 348:2): A mistake of a Nochri is permitted, e.g. to deceive him about a calculation or to uproot his loan, as long as he will not find out, and there is no Chilul Hash-m. Some forbid deceiving him, but if he himself erred, it is permitted.

Bach (2 DH Kosvu): Tosfos (113b DH Yachol) says in the name of the Ri and Aruch says that even though a Nochri's mistake is permitted, one may not deceive a Nochri who merely pretends not to realize.

SMA (6): The latter opinion learns from the fact that even Geneivas Da'as of a Nochri is forbidden.

Gra (CM 13): The Rambam learns from Rav Kahana that the Yisrael must say 'I rely on you.' The Rema omitted this. He learns from Shmuel and Ravina, and explains that Rav Kahana was Midas Chasidus.

Yam Shel Shlomo (10:18): Rashi explained that we seek a ruse to exempt the Yisrael, i.e. when the Nochri claimed from him. This is uprooting a debt. We do not seek a ruse to allow a Yisrael to collect from a Nochri. That is theft!

Pischei Teshuvah (4): Sha'ar Mishpat (2) says that if a Nochri gave money to Levi to buy merchandise and Levi profitted with it, even though l'Chatchilah it is no less than Geneivas Da'as and it is forbidden, he need not return it. Since Shlichus does not apply to Nochrim (Rema OC 450:6), it is as if Levi bought for himself and profitted from his own money.