Elliptic curve isogeny-based cryptographic scheme

A cryptographic correspondent device comprising a processor and a memory, the cryptographic correspondent device in communication with an other cryptographic correspondent device over a data communication system, the memory having stored thereon computer instructions which when executed by the processor cause the processor to implement a elliptic curve cryptographic scheme comprising:

• receiving, from the other cryptographic correspondent device, a first public key, the other cryptographic correspondent device having selected a first selected basis being either a first basis or a second basis, and the other cryptographic correspondent device having performed first key generation with respect to the first selected basis to generate a first private key and the first public key;
• determining if a linear combination, in a second selected basis being either the first basis or the second basis, is an identity point;
• performing second key generation with respect to the first basis if the scalar multiplication product is the identity point, otherwise performing second key generation with respect to the second basis, the second key generation generating a second private key and a second public key;
• communicating the second public key to the other cryptographic correspondent device;
• and
• generating a common key by combining the second private key and the first public key.

An elliptic curve cryptographic scheme performed between a pair of entities on computing devices communicating over a data communication system, the cryptographic scheme comprising:

• selecting, randomly, first private scalars in a first basis and second private scalars in a second basis;
• determining a first kernel in the first basis using the first private scalars and a second kernel in the second basis using the second private scalars;
• determining a first elliptic curve using the first kernel and a second elliptic curve using the second kernel;
• determining first auxiliary points based on a first isogeny of the first elliptic curve and second auxiliary points based on a second isogeny of the second elliptic curve;
• generating a private key comprising the first private scalars and the second private scalars;
• generating a public key comprising the first elliptic curve, the second elliptic curve, the first auxiliary points and the second auxiliary points;
• making available the public key to the other entity;
• receiving, from the other entity, an acquired public key, the acquired public key comprising a first-acquired elliptic curve, a second-acquired elliptic curve, first-acquired auxiliary points and second-acquired auxiliary points;
• and
• generating a common key, comprising:
• determining a first-consequent kernel using the first private scalars and the second-acquired auxiliary points;
• computing a second-consequent kernel using the second private scalars and the first-acquired auxiliary points;
• mapping a first-consequent elliptic curve from the first-acquired elliptic curve using the first-consequent kernel;
• mapping a second-consequent elliptic curve from the second-acquired elliptic curve using the second-consequent kernel;
• and
• determining the common key as the j-invariants of the first-consequent elliptic curve and the second-consequent elliptic curve.

A cryptographic correspondent device comprising a processor and a memory, the cryptographic correspondent device in communication with an other cryptographic correspondent device over a data communication system, the memory having stored thereon computer instructions which when executed by the processor cause the processor to implement a elliptic curve cryptographic scheme comprising:

• selecting, randomly, first private scalars in a first basis and second private scalars in a second basis;
• determining a first kernel in the first basis using the first private scalars and a second kernel in the second basis using the second private scalars;
• determining a first elliptic curve using the first kernel and a second elliptic curve using the second kernel;
• determining first auxiliary points based on a first isogeny of the first elliptic curve and second auxiliary points based on a second isogeny of the second elliptic curve;
• generating a private key comprising the first private scalars and the second private scalars;
• generating a public key comprising the first elliptic curve, the second elliptic curve, the first auxiliary points and the second auxiliary points;
• making available the public key to the other entity;
• receiving, from the other cryptographic correspondent device, an acquired public key, the acquired public key comprising a first-acquired elliptic curve, a second-acquired elliptic curve, first-acquired auxiliary points and second-acquired auxiliary points;
• and
• generating a common key, comprising:
• determining a first-consequent kernel using the first private scalars and the second-acquired auxiliary points;
• computing a second-consequent kernel using the second private scalars and the first-acquired auxiliary points;
• mapping a first-consequent elliptic curve from the first-acquired elliptic curve using the first-consequent kernel;
• mapping a second-consequent elliptic curve from the second-acquired elliptic curve using the second-consequent kernel;
• and
• determining the common key as the j-invariants of the first-consequent elliptic curve and the second-consequent elliptic curve.