<p>To characterize the permutation property of polynomials, complete permutation in multiplication (CPM) was introduced as a new concept in 2020. It has some applications in mathematics and cryptography. In this paper, we investigate the constructions of CPMs over finite fields with even characteristic. We construct six classes of CPMs <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x^{2^n-2-d}+a x^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo>-</mo> <mn>2</mn> <mo>-</mo> <mi>d</mi> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_{2^n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation>, which explain several sporadic experiment results. Moreover, we present some new constructions of CPMs from the known permutations, such as linearized polynomial, Dickson polynomial, permutation trinomial, and so on.</p>