EXHIBIT 32.1

CERTIFICATION OF CHIEF EXECUTIVE OFFICER

PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

I, Donald L. Ciffone Jr. certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that, to my knowledge, the Quarterly Report of Exar Corporation on Form 10-Q for the period ended June 30, 2004 fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 and that information contained in such Quarterly Report on Form 10-Q fairly presents in all material respects the financial condition and results of operations of Exar Corporation.

Date: August 9, 2004

/s/    DONALD L. CIFFONE JR.        

Donald L. Ciffone, Jr.

Chairman of the Board,

Chief Executive Officer and President

(Principal Executive Officer)