Under the federal perjury statutes, it is a crime for a witness testifying under oath before a grand jury to knowingly make any false material declaration. See 18 U.S.C. § 1623. The government must prove four elements beyond a reasonable doubt to obtain a conviction under Section 1623(a): the defendant testified under oath before a federal grand jury; the defendant’s testimony was false in one or more respects; the false testimony concerned matters that were material to the grand jury investigation; and the false testimony was knowingly given. United States v. Bridges, 717 F.2d 1444, 1449 n.30
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