p/2, '-y' if y < p/2) in this way when a tame T reaches a point Q = (x,y) and a wild W reaches a point Q' = -Q = (x,-y), this become a collision The tame and the wild will go on with 2 specular paths: Q -> Q + k*G -> Q + k*G -r*G ->  ... -> DP (x,y)    (x1,-y1)          (x2,y2)              (x0, y0) -Q -> -Q -k*G -> -Q - k*G +r*G -> ... -> DP (x,-y)  (x1,y1)         (x2, -y2)              (x0, -y0) until they reach 2 symetric DP, for both of which we know the private key. I have created a sort of "brownian motion", my kangaroos keep jumping back and forth. I'm not sure to fully understand your idea. You want to add a condition on the y parity to determine the jump ? You let the starting positions as they are or you translate them ? The start interval must be: [-(b-a)/2, +(b-a)/2] and yes, I want to add a condition on the y parity to determine the jump's direction. In this way a wild and a tame collide if they reach: 1) the same point 2) two symetric points In the interval [-(b-a)/2, +(b-a)/2], the x-coordinates of [1, +(b-a)/2] are equal to the x-coordinates of [-(b-a)/2, -1]; we should get more collisions, because we work in a space of 2^(n-1) points instead of 2^n. We introduce a equivalence class [P], where P ~ Q if Q = -P and the jump function goes from [P] to [P1], i.e. it is defined between 2 equivalence classes instead of between 2 points. A problem could be that many paths could enter in a loop before they reach a DP, because each kangaroo jumps back and forth and then a single kangaroo can collide with itself.
It is enough to modify slightly the way we generate the next jump: + jP[tamePosi.x % n]*G if y > p/2,  - jP[tamePosi.x % n]*G is y

p/2, '-y' if y < p/2) in this way when a tame T reaches a point Q = (x,y) and a wild W reaches a point Q' = -Q = (x,-y), this become a collision The tame and the wild will go on with 2 specular paths: Q -> Q + k*G -> Q + k*G -r*G ->  ... -> DP (x,y)    (x1,-y1)          (x2,y2)              (x0, y0) -Q -> -Q -k*G -> -Q - k*G +r*G -> ... -> DP (x,-y)  (x1,y1)         (x2, -y2)              (x0, -y0) until they reach 2 symetric DP, for both of which we know the private key. I have created a sort of "brownian motion", my kangaroos keep jumping back and forth. I'm not sure to fully understand your idea. You want to add a condition on the y parity to determine the jump ? You let the starting positions as they are or you translate them ? The start interval must be: [-(b-a)/2, +(b-a)/2] and yes, I want to add a condition on the y parity to determine the jump's direction. In this way a wild and a tame collide if they reach: 1) the same point 2) two symetric points In the interval [-(b-a)/2, +(b-a)/2], the x-coordinates of [1, +(b-a)/2] are equal to the x-coordinates of [-(b-a)/2, -1]; we should get more collisions, because we work in a space of 2^(n-1) points instead of 2^n. We introduce a equivalence class [P], where P ~ Q if Q = -P and the jump function goes from [P] to [P1], i.e. it is defined between 2 equivalence classes instead of between 2 points. A problem could be that many paths could enter in a loop before they reach a DP, because each kangaroo jumps back and forth and then a single kangaroo can collide with itself.

p/2, '-y' if y < p/2) in this way when a tame T reaches a point Q = (x,y) and a wild W reaches a point Q' = -Q = (x,-y), this become a collision The tame and the wild will go on with 2 specular paths: Q -> Q + k*G -> Q + k*G -r*G ->  ... -> DP (x,y)    (x1,-y1)          (x2,y2)              (x0, y0) -Q -> -Q -k*G -> -Q - k*G +r*G -> ... -> DP (x,-y)  (x1,y1)         (x2, -y2)              (x0, -y0) until they reach 2 symetric DP, for both of which we know the private key. I have created a sort of "brownian motion", my kangaroos keep jumping back and forth. I'm not sure to fully understand your idea. You want to add a condition on the y parity to determine the jump ? You let the starting positions as they are or you translate them ? The start interval must be: [-(b-a)/2, +(b-a)/2] and yes, I want to add a condition on the y parity to determine the jump's direction. In this way a wild and a tame collide if they reach: 1) the same point 2) two symetric points In the interval [-(b-a)/2, +(b-a)/2], the x-coordinates of [1, +(b-a)/2] are equal to the x-coordinates of [-(b-a)/2, -1]; we should get more collisions, because we work in a space of 2^(n-1) points instead of 2^n. We introduce a equivalence class [P], where P ~ Q if Q = -P and the jump function goes from [P] to [P1], i.e. it is defined between 2 equivalence classes instead of between 2 points. A problem could be that many paths could enter in a loop before they reach a DP, because each kangaroo jumps back and forth and then a single kangaroo can collide with itself.

p/2, '-y' if y < p/2) in this way when a tame T reaches a point Q = (x,y) and a wild W reaches a point Q' = -Q = (x,-y), this become a collision The tame and the wild will go on with 2 specular paths: Q -> Q + k*G -> Q + k*G -r*G ->  ... -> DP (x,y)    (x1,-y1)          (x2,y2)              (x0, y0) -Q -> -Q -k*G -> -Q - k*G +r*G -> ... -> DP (x,-y)  (x1,y1)         (x2, -y2)              (x0, -y0) until they reach 2 symetric DP, for both of which we know the private key. I have created a sort of "brownian motion", my kangaroos keep jumping back and forth. I'm not sure to fully understand your idea. You want to add a condition on the y parity to determine the jump ? You let the starting positions as they are or you translate them ? The start interval must be: [-(b-a)/2, +(b-a)/2] and yes, I want to add a condition on the y parity to determine the jump's direction. In this way a wild and a tame collide if they reach: 1) the same point 2) two symetric points In the interval [-(b-a)/2, +(b-a)/2], the x-coordinates of [1, +(b-a)/2] are equal to the x-coordinates of [-(b-a)/2, -1]; we should get more collisions, because we work in a space of 2^(n-1) points instead of 2^n. We introduce a equivalence class [P], where P ~ Q if Q = -P and the jump function goes from [P] to [P1], i.e. it is defined between 2 equivalence classes instead of between 2 points. A problem could be that many paths could enter in a loop before they reach a DP, because each kangaroo jumps back and forth and then a single kangaroo can collide with itself.