i think all DPs from previous work will be in the left 1/32 part of interval(in the begining).
If you just multiply DPs by 32 then they will be just points not DPs becouse they will not have leading zeros.
If i wrong can you explaine in very small range how transform DP by 32 times and do not loose leading zeros? Thanks.
No, you don't transform the old DPs, you change only their private keys (changing the generator G). The same for the old jumps. The only point you change is P, the public key for which you are looking the private key.
Instead of multiplying the old points by 32 you divide (= multiply by inv(32)) the new public point P -> P'=inv(32)*P.
If P lies in the [1, 2^119 - 1] interval, then P' lies in the
[1/32, ...., 31/32, 1, 33/32, ....., 2, ......3, .......4, ................., 2^114 -1/32] interval.
Note that the private keys of the old DPs are 'integer'.
In this way you spread the old DPs uniformly in a set x32 bigger. This interval contains all the previous points [1,2,3,..., 2^114-1],
but not at the beginning! You can change now the private keys changing the generator, G -> G' = inv(32)*G:
[1, .........,31, 32, 33, .........2*32,..........3*32, .............., 4*32,.................32*2^114 - 1]
the private keys of the old DPs have become all multiplies of 32.
If for example a DP has private key = 78543 respect of G, it has private key = 32*78543 respect of G', the point remains the same (same x-coordinate), you have changed only the point G.
The private key of P' respect of G' is the same private key of P respect of G.
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Example:
first interval = [1,2,3,4,5,6,7] (3 bit)
DPs = 2*G, 5*G
second interval = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,....,27,28,29,30,31] (5 bit)
you want to find P = 11*G, you know only that the P lies in this interval
instead of searching P in the second interval [1*G, 2*G, ...., 31*G]
you search P'=inv(4)*P (a different point) in a
third interval [1*G', 2*G', ......, 31*G'] (that contains the first interval but not at the beginning + P')
The old DPs are now: 8*G' and 20*G' (same points, different private keys). All the points of the first interval are now in the third one too.
The private key of P' = 11, in fact 11*G' = 11*inv(4)*G --> P = 4*P' = 4*11*inv(4)*G = 11*G