  An analysis is made of the properties and conditions for the existence of 3-
and 5-isogenies of complete and quadratic supersingular Edwards curves. For the
encapsulation of keys based on the SIDH algorithm, it is proposed to use
isogeny of minimal odd degrees 3 and 5, which allows bypassing the problem of
singular points of the 2nd and 4th orders, characteristic of 2-isogenies. A
review of the main properties of the classes of complete, quadratic, and
twisted Edwards curves over a simple field is given. Equations for the isogeny
of odd degrees are reduced to a form adapted to curves in the form of
Weierstrass. To do this, use the modified law of addition of curve points in
the generalized Edwards form, which preserves the horizontal symmetry of the
curve return points. Examples of the calculation of 3- and 5-isogenies of
complete Edwards supersingular curves over small simple fields are given, and
the properties of the isogeny composition for their calculation with
large-order kernels are discussed. Equations are obtained for upper complexity
estimates for computing isogeny of odd degrees 3 and 5 in the classes of
complete and quadratic Edwards curves in projective coordinates; algorithms are
constructed for calculating 3- and 5-isogenies of Edwards curves with
complexity 6M + 4S and 12M + 5S, respectively. The conditions for the existence
of supersingular complete and quadratic Edwards curves of order 4x3mx5n and
8x3mx5n are found. Some parameters of the cryptosystem are determined when
implementing the SIDH algorithm at the level of quantum security of 128 bits.
