Computing isogenies between elliptic curves is a significant part of post-quantum cryptography with many practical applications (for example, in SIDH, SIKE, B-SIDH, or CSIDH algorithms). Comparing to other post-quantum algorithms, the main advantages of these protocols are smaller keys, the similar idea as in the ECDH, and a large basis of expertise about elliptic curves. The main disadvantage of the isogeny-based cryptosystems is their computational efficiency - they are slower than other post-quantum algorithms (e.g., lattice-based). That is why so much effort has been put into improving the hitherto known methods of computing isogenies between elliptic curves. In this paper, we present new formulas for computing isogenies between elliptic curves in the extended Jacobi quartic form with two methods: by transforming such curves into the short Weierstrass model, computing an isogeny in this form and then transforming back into an initial model or by computing an isogeny directly between two extended Jacobi quartics.

The concept of a hybrid scheme with connection of SIDH and ECDH is nowadays very popular. In hardware implementations it is convenient to use a classical key exchange algorithm, which is based on the same finite field as SIDH. Most frequently used hybrid scheme is SIDH-ECDH. On the other hand, using the same field as in SIDH, one can construct schemes over Fpn, like Diffie-Hellman or XTR scheme, whose security is based on the discrete logarithm problem. In this paper, idea of such schemes will be presented. The security of schemes, which are based on the discrete logarithm problem over fields Fp; Fp2 ; Fp4 ; Fp6 and Fp8 , for primes p used in SIDH, will be analyzed. At the end, the propositions of practical applications of these schemes will be presented.