
609 F.3d 495 (2010)
UNITED STATES of America, Appellee,
v.
Arcadio RAMIREZ, José Luis Rodriguez, Appellants-Defendants,
Alex Luna, Defendant.
Docket Nos. 07-2912-cr(L), 08-6210-cr(CON).
United States Court of Appeals, Second Circuit.
Argued: December 17, 2009.
Decided: June 29, 2010.
*497 Robert J. Sullivan, Jr., Westport, CT, for Defendant-Appellant Rodriguez.
Barry A. Weinstein, Goldstein & Weinstein, Bronx, NY, for Defendant-Appellant Ramirez.
Harold H. Chen, Assistant United States Attorney (William J. Nardini, Assistant United States Attorney, on the brief), for Nora R. Dannehy, United States Attorney, District of Connecticut, New Haven, CT, for Appellee United States of America.
Before: CALABRESI, CABRANES, and B.D. PARKER, Circuit Judges.
B.D. PARKER, JR., Circuit Judge:
This appeal stems from the defendants' convictions in the United States District Court for the District of Connecticut (Underhill, J.) for their participation in a large-scale cocaine trafficking conspiracy. The indictment charged Rodriguez with being a major supplier of cocaine to a Danbury, Connecticut drug distribution network from 2002 to 2005, and charged Ramirez with similar conduct from 1998 to 2005. Ultimately, Rodriguez was convicted of conspiracy to distribute 5 kilograms of cocaine or 50 grams of cocaine base, and Ramirez for 500 grams of cocaine or 5 grams of cocaine base. See 21 U.S.C. § § 841(a)(1), 841(b)(1), 846.
On appeal, Rodriguez challenges his conviction on two separate grounds, while Ramirez challenges his sentence. Rodriguez argues that two rebuttal witnesses police officer Waldo Cuba and Maria Roblesshould not have been permitted to testify about collateral matters, and that their testimony severely prejudiced his defense. He also complains that the bill of particulars filed by the government on the eve of trial was inadequate, and that the indictment should have been dismissed. Ramirez, for his part, claims that the sentencing judge improperly found him responsible for 15 kilograms of cocaine, resulting in a 204-month sentence, after the jury expressly rejected such a large drug quantity in its special interrogatory responses. Thus, according to Ramirez, the district court improperly relied on "acquitted conduct" to establish drug quantity at sentencing.
We find that the district court erred in admitting Officer Cuba's testimony to impeach Rodriguez. Nonetheless, we affirm his conviction, ultimately finding this error harmless. Because we reject Rodriguez's other claims, as well as Ramirez's sentencing challenge, the defendants' convictions and sentences are affirmed.

I. BACKGROUND
The defendants were charged and convicted for their roles in a drug distribution conspiracy involving the weekly transport of cocaine from Brooklyn, New York to Danbury, Connecticut. Co-conspirator José Adames (also known as "Pompa" or "Ponpa") led the trafficking operation. From 2002 to 2005, Rodriguez was alleged to be Adames's driver for many of the trips to Danbury, where the drugs were purchased for resale by Alex Luna, another co-conspirator. Ramirez, in his role, also often accompanied Adames to Connecticut, and was responsible for supplying *498 and distributing cocaine in Brooklyn over an even longer period.
Rodriguez was arrested in Massachusetts and, according to the testimony of one of the arresting officers, confessed to his role in the conspiracy while being transported back to Connecticut for prosecution. At trial, the government also offered the testimony of three other co-conspirators who described Rodriguez's involvement with Adames, Luna, and Ramirez in regular cocaine transactions. Finally, the government presented video surveillance of Rodriguez meeting with Adames and Luna in Danbury, although the video did not show any drugs changing hands.
Rodriguez testified in his own defense. He foreswore any knowing involvement in the narcotics conspiracy and denied the alleged confession. Instead, he claimed that, although he had acted as Adames's driver over a period of years, he had no knowledge of any drug transactions. According to Rodriguez, he came to know Adames socially, through friends and family; when Rodriguez later lost his job, Adames began to ask Rodriguez to drive him places, including Danbury, to visit relatives. Rodriguez testified that he never saw or knew of any cocaine on these trips. Because he was without work, Rodriguez says he continued to chauffeur Adames periodically until late 2004 when Rodriguez's mother warned him that Adames was thought to be involved with drugs. Rodriguez claims that he confronted Adames about these rumors and, although they were denied, that he subsequently distanced himself from Adames.
The government sought to undermine Rodriguez's defense by, among other things, cross-examining him about his contact with drugs during this period. Relying on Rodriguez's denial of any involvement with cocaine on both direct and cross-examination, the government also sought to introduce the rebuttal testimony of police officer Waldo Cuba and Alex Luna's girlfriend, Maria Robles. In particular, Officer Cuba reported seeing Rodriguez handling cocaine during an unrelated drug stop shortly after the conspiracy had ended. Robles, for her part, testified about Rodriguez's involvement with drug trafficking during the indicted period. The trial court acknowledged that this testimony had the potential to be extremely damaging; nonetheless, over Rodriguez's objections, it admitted the testimony of both witnesses.
Ultimately, the jury convicted each defendant of conspiracy to possess with intent to distribute cocaine, in violation of 21 U.S.C. §§ 841, 846. Rodriguez was subsequently sentenced to 120 months imprisonment, while Ramirez was sentenced to 204 months imprisonment.

II. RODRIGUEZ

A. Admission of impeachment testimony
Rodriguez claims that the district court erred when it permitted the rebuttal testimony of Cuba and Robles. The testimony of both witnesses was offered to prove that Rodriguez had had previous contact with drugs, thereby impeaching his statements to the contrary as well as his overall credibility. Officer Cuba provided testimony describing Rodriguez's involvement in a separate drug-related stop in Queens on March 30, 2005. Importantly, this stop occurred after the period of the charged conspiracy and, in fact, after Adames had already been arrested. Cuba testified that, while posted with the Special Narcotics Enforcement Unit, he had seen Rodriguez outside a vehicle placing what appeared to be plastic bags of cocaine into a *499 small plastic box. Cuba subsequently followed the vehicle and radioed his fellow officers, who pulled Rodriguez over and arrested the SUV's two occupants. The arrest report indicates that Rodriguez later received an adjournment in contemplation of dismissal ("ACD") and that all records relating to the incident were sealed. In view of this fact, the prosecution was unable to account for how it had received a copy of the sealed arrest report, claiming that it mysteriously arrived by fax with no identifiable source.
In her testimony, Maria Robles, briefly described a conversation with Rodriguez while the conspiracy was ongoing, in which he recounted swallowing "balloons" of cocaine in order to smuggle them into the country from the Dominican Republic. She also reported seeing Rodriguez accompany Adames on drug deliveries to Danbury on a weekly basis, and stated that Rodriguez had made deliveries alone on two or three occasions.

1. Standard of review
In general, we will not overturn the district court's decision to admit or reject evidence absent an abuse of discretion. See United States v. Esdaille, 769 F.2d 104, 108 (2d Cir.1985). Even if we conclude that the admission of the rebuttal testimony here was in error, the government contends that any such error was harmless. See Fed.R.Crim.P. 52(a); United States v. Madori, 419 F.3d 159, 168 (2d Cir.2005). Broadly speaking, the erroneous admission of evidence is harmless where it "had no substantial and injurious effect or influence on the jury verdict," judged in relation to the total evidence on the issue in question. United States v. Garcia, 413 F.3d 201, 217 (2d Cir.2005) (quotation marks omitted).

2. Analysis
Generally, the Federal Rules of Evidence bar the admission of extrinsic evidence, like Officer Cuba's testimony, related to the past conduct of a witness. The government concedes as much, acknowledging that the testimony related to an entirely collateral matter and did not serve as proof of the charged offense. Indeed, because Rodriguez's March 30, 2005 arrest occurred after the end of the indicted conspiracy, Officer Cuba's rebuttal testimony would normally have been precluded by Federal Rule of Evidence 608(b), which prohibits extrinsic evidence of exactly this kind. Nonetheless, the government argues that this evidence was admissible for the purpose of "impeachment by contradiction," which operates as a limited exception to Rule 608(b). "Where a defendant testifies on direct about a specific fact, the prosecution is entitled to prove... that he lied as to that fact." United States v. Beverly, 5 F.3d 633, 639-40 (2d Cir.1993) (permitting use of extrinsic evidence). More precisely, this doctrine provides that when a witness puts certain facts at issue in his testimony, the government may seek to rebut those facts, including by resorting to extrinsic evidence if necessary. According to the government, the point of this exception is that a defendant may not invoke the Federal Rules of Evidence in order to shield his perjury from contradiction.
The government contends that Rodriguez's statements on both direct and cross-examination triggered the exception, but the trial record simply does not bear out this rationale. It is an open question in our Court whether the government can present extrinsic evidence to impeach by contradiction a statement made by the defendant on cross-examination, where such evidence would otherwise be barred by the *500 Federal Rules of Evidence.[1] We need not resolve this question because, as a factual matter, Officer Cuba did not impeach either Rodriguez's testimony on direct examination or his responses on cross-examination.
By highlighting Rodriguez's unrelated stop while in possession of cocaine, Officer Cuba's rebuttal testimony was purportedly offered to impeach Rodriguez's statements that he had never seen or handled drugs. Yet Rodriguez never gave the testimony that the government ascribes to him. The transcript is pellucid that on both direct and cross-examination, Rodriguez was responding to a series of questions about his work for Adames and the time-period covered by the alleged conspiracy. For instance, at the end of a lengthy colloquy focused on Adames's activities from 2001 to 2004, defense counsel asked Rodriguez:
Def. Counsel: During the time that you were just describing, did you see any deliveries of drugs of any kind?
Rodriguez: No, I never see no drugs.
13 Gov't App'x at 118-19 (emphasis added). The government pins its rebuttal claim on this testimony, yet it is abundantly clear from the context that Rodriguez was not issuing a blanket denial of ever having seen drugs.
The very same is true of Rodriguez's cross-examination testimony, where he was asked about his activities during the charged conspiracy:
Gov't: Mr. Rodriguez, you're claiming you never saw any drugs in this whole time period, correct?
Rodriguez: No, never.
Gov't: Except for marijuana. You saw marijuana, right?
Rodriguez: Yes.
Gov't: Okay, but you never saw cocaine?
Rodriguez: No.
Gov't App'x at 134 (emphasis added). The government points to no instances where Rodriguez forswore, as a universal matter, ever having seen drugs of any kind. Compare Walder v. United States, 347 U.S. 62, 65, 74 S.Ct. 354, 98 L.Ed. 503 (1954). Nor did his reference to having had "a little problem before" provide a predicate for admitting Officer Cuba's testimony about the arrest. The government contends that this "little problem" referred to the drug stop, which opened the door to its rebuttal testimony. But reading Rodriguez's statement in context, again, makes clear that he was referring to an earlier eventas he allegedly decided to distance himself from Adamesnot an arrest that occurred after the conspiracy's conclusion: "[Y]ou know, I don't want to get no more trouble, so I *501 think Mom was right at that time." Most of all, none of these statements show that Rodriguez staked his credibility before the jury on any expansive assertion about lifelong avoidance of drugs. Indeed, the government inaccurately characterizes the trial record when it suggests as much.
As a result, we see no basis for admitting Officer Cuba's testimony under the "impeachment by contradiction" doctrine and conclude that it should have been barred by Federal Rule of Evidence 608(b). On the other hand, we sustain the district court's admission of Maria Robles's testimony. That testimonydescribing Rodriguez's transport and delivery of cocainerelated to the charged conspiracy, and contradicted Rodriguez's statement on direct examination that he had no contact with drugs during the relevant period. We cannot say that the admission of this evidence was an abuse of discretion.

3. Harmless Error
"A district court's erroneous admission of evidence is harmless if the appellate court can conclude with fair assurance that the evidence did not substantially influence the jury." United States v. Al-Moayad, 545 F.3d 139, 164 (2d Cir.2008) (internal quotation marks omitted). In reviewing for harmless error, "we principally consider (1) the overall strength of the prosecution's case; (2) the prosecutor's conduct with respect to the improperly admitted evidence; (3) the importance of the wrongly admitted [evidence]; and (4) whether such evidence was cumulative of other properly admitted evidence." United States v. Kaplan, 490 F.3d 110, 123 (2d Cir.2007) (internal quotation marks omitted). We have frequently stated that the strength of the government's case is the most critical factor in assessing whether error was harmless. See, e.g., United States v. Lombardozzi, 491 F.3d 61, 76 (2d Cir.2007).
There is no question that a police officer's testimony about a previous unrelated drug stop can prove extremely damaging to a defendant at trial. It functions essentially as evidence of criminal propensity and presents exactly the risk of undue prejudice that the Federal Rules of Evidence are designed to guard against. We do not treat the erroneous admission of such evidence lightly, and we weigh its effect on the jury carefully. Cf. United States v. McCallum, 584 F.3d 471, 478 (2d Cir.2009) ("In a different case, in which... the government's other evidence was not overwhelming, or where the other harmless error factors tilted more strongly in the defendant's favor, or where the government's summation emphasized the prior [misconduct], a different result could well be indicated.").
Even with these precautions, we conclude that the government's evidence was sufficiently conclusive and its case sufficiently robust such that the error in Rodriguez's case was harmless. The government introduced a series of co-conspirators who testified to Rodriguez's knowing involvement in the cocaine trafficking operation. Nelson Rosa testified about cocaine deliveries Rodriguez made to him in Danbury, about money that Rodriguez received from Alex Luna in exchange for drugs, about how Rodriguez would retrieve cocaine from the vehicle he drove for Adames, and about how Rodriguez helped Adames cook cocaine into crack.
José Pena provided similar testimony, describing Rodriguez's role as a driver who would often "bring down the merchandise," and who was frequently present when Adames discussed cocaine sales or handed off drugs hidden in the car's secret compartment. Nicky Carrasquillo recounted for the jury how Rodriguez accompanied *502 Adames on repeated drug deliveries, and how he too had witnessed Rodriguez and Adames removing drugs from secret compartments in two vehicles. He also described seeing Rodriguez "compressing" cocaine at Adames's housethat is, mixing relatively pure cocaine with other substances in order to increase the overall resale profits. Finally, Maria Robles, another cooperating co-conspirator, testified to seeing Rodriguez and Adames deliver cocaine, approximately one kilogram at a time, to her boyfriend Alex Luna on a weekly basis. She also told the jury that Rodriguez and Adames once ran Luna's drug business in Danbury while she and Luna vacationed in the Dominican Republic.
In addition to these witnesses, the prosecution called Special Agent Rodney George, who described statements Rodriguez made shortly after his arrest. According to this testimony, Rodriguez confessed to his involvement in Adames's drug trafficking operation while he was transported from Boston, where he was arrested, to Connecticut. As he rode with Agent George, Rodriguez allegedly told him how he had become aware of Adames's cocaine trafficking, had met Adames's supplier, and had driven Adames to make frequent deliveries in Danbury.
Lastly, the government produced video surveillance of Rodriguez and Adames meeting Luna in Danbury. This surveillance showed the three men arriving in a parking lot, warily surveying their surroundings, and then proceeding into an apartment. While the footage, taken outside, does not show any drugs changing hands, the meeting followed a phone call in which Luna asked Adames to deliver 200 grams of cocaine.
Together, all this evidence is at odds with Rodriguez's claim that, although he frequently drove Adames to Danbury for a period of years, he knew nothing of Adames's drug trafficking activities. Needless to say, given the scale and frequency of Adames's cocaine deliveries to Luna, such a defense strains belief. Based on the strength of the government's evidence to the contrary, and the many witnesses who testified to Rodriguez's knowing involvement, we conclude that the erroneous admission of Agent Cuba's rebuttal testimony was harmless.

B. Bill of particulars
We conclude that the district court did not err in determining that the government's submission in response to the district court's order for a bill of particulars was sufficient. The district court granted Rodriguez's motion for a bill of particulars, and in response, the government submitted a 62-page summary of the evidence that it intended to introduce at trial. Although the government's submission did not take the traditional format of a bill of particulars, the district court found that the government's submission contained sufficient specificity to inform Rodriguez of the charges against him and to prevent undue surprise at trial. In light of the fact that Rodriguez has not, on appeal, identified "any particular charge on which further detail would have been helpful," In re Terrorist Bombings of U.S. Embassies in E. Afr., 552 F.3d 93, 151 (2d Cir.2008), we find no error in the district court's ruling, and we will not overturn Rodriguez's conviction on the ground that the government's submission was not formatted as a traditional bill of particulars.
Whether to grant a bill of particulars is generally a decision entrusted to the sound discretion of the district court. See United States v. Barnes, 158 F.3d 662, 665-66 (2d Cir.1998). That being the case, a district court's determination that a bill of particulars is sufficient would appear to be entitled to an even greater measure of *503 deference. See id. at 665 ("A district court judge ... has the discretion to deny a bill of particulars if the information sought by defendant is provided in the indictment or in some acceptable alternate form." (quotation marks omitted)).
In this case, the bill of particulars, though lengthy, significantly condensed the voluminous discovery produced by the prosecution into a form that apprised Rodriguez of what the government would seek to prove at trial. See United States v. Davidoff, 845 F.2d 1151, 1154 (2d Cir.1988) (A bill of particulars enables a "defendant to prepare for trial, to prevent surprise, and to interpose a plea of double jeopardy should he be prosecuted a second time for the same offense." (quotation marks omitted)). The district court made clear that it ordered a bill of particulars because so much discovery was produced to the defendants, not too little. That being the case, the bill, though not formatted in the traditional fashion, adequately and appropriately explicated the charges Rodriguez faced so that he could defend against them. He cannot show that he was entitled to anything more.
We have considered Rodriguez's remaining arguments on appeal and find them meritless. We therefore affirm Rodriguez's conviction.

III. RAMIREZ
Ramirez appeals arguing that the district court erred by basing his sentence on a finding that he was responsible for at least 15 kilograms of cocaine. When a district court makes a finding of fact with respect to the amount of drugs attributable to a defendant, we review that finding for clear error. United States v. Hazut, 140 F.3d 187, 190 (2d Cir.1998); see also United States v. Cavera, 550 F.3d 180, 190 (2d Cir.2008) (en banc) ("A district court commits procedural error [at sentencing] where it ... rests its sentence on a clearly erroneous finding of fact."). "The clearly erroneous standard requires us to uphold the ruling of the court below unless we are `left with the definite and firm conviction that a mistake has been committed.'" United States v. Reilly, 76 F.3d 1271, 1276 (2d Cir.1996) (quoting United States v. Gypsum Co., 333 U.S. 364, 395, 68 S.Ct. 525, 92 L.Ed. 746 (1948)).
Having reviewed the record, we conclude that there was ample evidence at trial, at sentencing, and in the PSR to support the district court's finding that Ramirez conspired to distribute (or conspired to possess with intent to distribute) an amount of cocaine that was "very easily over the 15-kilogram minimum." We conclude, therefore, that the district court did not commit clear error. We have considered Ramirez's additional arguments on appeal and find them meritless. We affirm Ramirez's sentence.

IV. CONCLUSION
For the foregoing reasons, the defendants' convictions and sentences are AFFIRMED.
NOTES
[1]  We have previously held that a defendant's statements on cross-examination may be impeached by evidence otherwise suppressed under the exclusionary rule. See United States v. Atherton, 936 F.2d 728, 734 (2d Cir. 1991) (citing United States v. Havens, 446 U.S. 620, 100 S.Ct. 1912, 64 L.Ed.2d 559 (1980)); Beverly, 5 F.3d at 639-40 (stating, in dicta, that "[t]he same holds true for defendant's false statements on cross-examination"). However, like other courts, we have been reluctant to extend this principle to evidence prohibited by the Federal Rules of Evidence, which often serve different policy goals. See United States v. Lawson, 683 F.2d 688, 692 (2d Cir. 1982) (distinguishing Havens because "[a] principal purpose of the exclusionary rule under Miranda is to deter police officers while Rules 410 and 11(e)(6) are designed to encourage plea bargaining"); see also Walder v. United States, 347 U.S. 62, 66, 74 S.Ct. 354, 98 L.Ed. 503 (1954) (prohibiting impeachment evidence where the government had "smuggle[d] in" the impeaching opportunity in the course of cross-examination) (citing Agnello v. United States, 269 U.S. 20, 35, 46 S.Ct. 4, 70 L.Ed. 145 (1925)); United States v. Pantone, 609 F.2d 675, 683 (3d Cir. 1979); United States v. Warledo, 557 F.2d 721, 726 (10th Cir.1977); United States v. Lambert, 463 F.2d 552, 557 (7th Cir. 1972).
