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I am trying to read a paper in cryptography. In key generation phase, paper give a definition for bilinear like G and Gt be two cyclic groups of prime order p

$e: G * G \to G_t$. be a map with the following properties: enter image description here

and in one place i found a definition like enter image description here

my doubt is in the last part. Specifically in m.e(g1,gn)^t suppose m is any message, g1=3,gn=13 t=6 (for more information v=5,gi=10.no problem whether all the value assumptions are true or not). then how can i compute e(g1,gn)^t part? sorry for my bad math notation

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  • $\begingroup$ You need Miller algorithm to calculate the map, and it could be a challenge to follow properties of rational functions on the curve with variables from field extensions. crypto.stanford.edu/miller $\endgroup$ – Vadym Fedyukovych Dec 3 '15 at 12:40
  • $\begingroup$ mathoverflow.net/questions/68942/… $\endgroup$ – Vadym Fedyukovych Dec 3 '15 at 14:30
  • $\begingroup$ Sorry, still i didnt get that. It will be great if you explain me with my given sample data.i got the first two part in cipher text. Remaining is e(g1,gn)^t. @Vadym Fedyukovych $\endgroup$ – John Mathew Dec 3 '15 at 15:41
  • $\begingroup$ The bitmap in the question is from this paper: Cheng-Kang Chu, Sherman S.M. Chow, Wen-Guey Tzeng, Jianying Zhou, Robert H. Deng, Key-Aggregate Cryptosystem for Scalable Data Sharing in Cloud Storage in IEEE Transactions on Parallel and Distributed Systems, vol. 25, no. 2,(Feb., 2014). $\endgroup$ – fgrieu Dec 3 '15 at 17:49
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    $\begingroup$ I was trying to give an estimate on problem complexity. It will be hard. To understand bilinear map, one need to start from finite fields textbook up to whatever it takes to learn Miller algorithm. If you prefer "data sharing implementation: done" path, consider a library with map implemented there. For example, PBC, introduced at the thesis referenced in the other question. @John Mathew $\endgroup$ – Vadym Fedyukovych Dec 3 '15 at 19:57
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According to PBC manual, Pairing functions, map $\hat e(,)$ (without t-power) could be calculated as follows:

pairing_pp_t pp;
pairing_pp_init(pp, g1, pairing);
pairing_pp_apply(r1, gn, pp); // r1 = e(g1, gn)
pairing_pp_clear(pp);

Before calculating the map, pairing parameters must be initialized.

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  • $\begingroup$ you did in program. But can you tell on example did by hand. I mean with numerical value?. Some where i found that e(3,11)=3 mod 11. Is it correct. if not can you explain with this kind of example? sorry for i am being a disturbance to you :P keep in mind that my exact problem is e(g1,gn)^t @Vadym Fedyukovych $\endgroup$ – John Mathew Dec 7 '15 at 4:30
  • $\begingroup$ printf(r1); @John Mathew $\endgroup$ – Vadym Fedyukovych Jan 3 '16 at 10:36

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