tower of extension field while working on tate pairing, i have to implement towering technique. like i have point p on F(q) and point Q(F(q^k)) (here embedding degree k=12 for BN curve).

instead of taking a point Q on F(q^12), i take this point on twisted curve E'(q^2).

finally i need to map this point to F(q^12),how can i use this towering technique (2-2-3), because output of tate pairing for BN curve should be F(q^12).
i have studied different articles but still confused, any numerical example would be very helpful.
thanks in advance
Are you sure you want 2-2-3, not 2-3-2? The latter is more common in my experience.

None the less, the first step is almost always done with a quadratic extension where i^2=-1. So you just need to implement complex numbers over your field of choice.

Multiplying complex numbers is simple if you just treat them as (x+iy) - I hope that's basic.

For inversion see See "Implementing cryptographic pairings", M. Scott, section 3.2. (ftp://136.206.11.249/pub/crypto/pairings.pdf)

And here's my code for this step: https://code.google.com/p/go/source/browse/bn256/gfp2.go?repo=crypto

For the cubic extension see http://eprint.iacr.org/2006/471.pdf. I used τ^3=i+3. The multiplication is little more complex, but still just an application of basic algebra. For inversion, I'll have to point you at the comments that I wrote in my code: https://code.google.com/p/go/source/browse/bn256/gfp6.go?repo=crypto#242

Now you're at q^6 you can work in the twisted curve.

Lastly, the quadratic extension to q^12 is similar to the original extension. After the cubic, it should be easy! I used ω²=τ because I was following http://cryptojedi.org/papers/dclxvi-20100714.pdf.

For exactly how you get from the twist over q^6 to the curve over q^12 depends on the exact homomorphism in question.