The following context is based on elliptic curves in short-weierstrass form y^2 = x^3 + b.
I know that elements of a non-prime order cyclic group G can be moved to its subgroup H by a process called "cofactor clearing". You just have to simply multiply a cofactor by an element of main group that ends up giving an element in its subgroup. Example given on- Why such a complicated way of cofactor clearing?
I want to know if there's a way/formula/algorithm for we can do the things in opposite, i.e., "moving elements from cyclic subgroup to its cyclic parent group" such that we get elements which belong to the parent group (respecting group laws ofc). I have tried taking (cofactor^-1) but didn't work.
NOTE: homomorphism should not be used as it would map the element of subgroup order to an element of same subgroup order. Here we want element of parent group order.
If someone can help me with this (with a good example) would be really appreciated...
EDIT: Let me clarify what I am looking for in a simple terms, pls read carefully - I need a formula that can be applied on any curve, say for e.g. y^2 = x^3 + 2 with prime = 157, parent group order = 172 and subgroups = 4 x 43. Suppose I have a random generator point "A" (on subgroup 43) and multiply it with some unknown scalar k that gives point "B" (on subgroup 43). So how should I get resulting points A' and B' on parent group G having order 172 keeping the scalar k preserved (and unknown)? A SAGE code would be helpful. You can think of a mapping of two points A and B H→G where all properties are preserved along with scalar mult. but the resulting points A' and B' have to belong to the parent group G (which has order 172) respecting scalar k.