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We propose a design principle of digital logic gates based on Josephson cryotrons. The operation mode of digital logic gates AND and OR based on tunnel junctions 'superconductor-insulator-superconductor' was described. Mathematical models of the switching processes in the logic elements were developed, the transient characteristics during the logic state change were calculated, and the basic parameters of these models were determined. It was shown that digital logic gates 'AND' and 'OR' can be realized based on individual Josephson cryotrons and their logic state can be controlled by using current pulses as control inputs. Such logic gates satisfy all the requirements for digital logic elements and have switching time of about 2-3 ps. [ABSTRACT FROM AUTHOR]