# US 2018/0323973 A1 · Elliptic curve isogeny-based cryptographic scheme > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 2018/0323973 A1 - Document ID: US20180323973A1 - Status: Published application - Kind: A1 - Application number: 15/800,813 - Filed: 2017-11-01 - Published: 2018-11-08 - Earliest claimed: 2016-11-07 - Assignees: Infosec Global Inc. - Applicants: Infosec Global Inc. - Inventors: Vladimir SOUKHAREV - Web page: https://patentgrep.com/patent/US20180323973A1 ## CPC - H04L 9/3066: {involving algebraic varieties, e.g. elliptic or hyper-elliptic curves} - G06F 7/725: {over elliptic curves} - H04L 9/0825: {using asymmetric-key encryption or public key infrastructure [PKI], e.g. key signature or public key certificates} - H04L: TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION (arrangements common to telegraphic and telephonic communication H04M) - G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N) ## Abstract Elliptic curve cryptographic schemes performed between a pair of cryptographic correspondent computing devices. In an aspect, a first entity generates a first basis point in a first selected basis being, either a first basis (A) or a second basis (B), and performs a first key generation in the selected basis. A second entity receives the public key and determines the product of a predetermined scalar in a second selected basis being either the first basis (A) or the second basis (B) and one of the first auxiliary points. If the product is an identity point, performs second key generation in the second selected basis, otherwise performing second key generation in either of the first basis (A) or the second basis (B). A common key is generated using the private keys and public keys. In another aspect, a scheme is performed symmetrically between two entities to generate a common key. ## Claims ### Claim 1 An elliptic curve cryptographic scheme performed between a first entity on a first computing device and a second entity on a second computing device communicating over a data communication system, the cryptographic scheme comprising: performing, by the first entity: selecting a first selected basis being either a first basis or a second basis; performing first key generation with respect to the first selected basis to generate a first private key and a first public key; and communicating the first public key to the second entity; performing, by the second entity: 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; and communicating a second public key to the second entity; performing, by the first entity: generating a common key by combining the first private key and the second public key; and performing, by the second entity: generating a common key by combining the second private key and the first public key. ### Claim 2 (depends on 1) The elliptic curve cryptographic scheme of claim 1, wherein first key generation comprises: generating a first private key; and generating a cryptographic corresponding first public key comprising a first elliptic curve and first auxiliary points; and wherein second key generation comprises: generating a second private key; and generating a cryptographic corresponding second public key comprising a second elliptic curve and second auxiliary points. ### Claim 3 (depends on 2) The elliptic curve cryptographic scheme of claim 2, wherein determining the linear combination comprises determining a scalar multiplication product of a predetermined scalar in the second selected basis and one of the first auxiliary points. ### Claim 4 (depends on 2) The elliptic curve cryptographic scheme of claim 2, wherein determining the linear combination comprises determining a scalar multiplication product of a predetermined scalar in the first selected basis and one of the first auxiliary points. ### Claim 5 (depends on 2) The elliptic curve cryptographic scheme of claim 2, wherein determining the linear combination comprises determining a scalar multiplication product of a predetermined scalar in the first selected basis, a predetermined scalar in the second selected basis, and one of the first auxiliary points. ### Claim 6 (depends on 5) The elliptic curve cryptographic scheme of claim 5, further comprising determining whether the values of the first public key are in the right domain by determining that such values are in defined torsion subgroups. ### Claim 7 (depends on 2) The elliptic curve cryptographic scheme of claim 2, wherein determining the linear combination comprises determining a sum of a first private scalar multiplied by the first auxiliary point and a second private scalar multiplied by the second auxiliary point. ### Claim 8 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. ### Claim 9 (depends on 8) The cryptographic correspondent device of claim 8, wherein first key generation comprises: generating a first private key; and generating a cryptographic corresponding first public key comprising a first elliptic curve and first auxiliary points; and wherein second key generation comprises: generating a second private key; and generating a cryptographic corresponding second public key comprising a second elliptic curve and second auxiliary points. ### Claim 10 (depends on 9) The cryptographic correspondent device of claim 9, wherein determining the linear combination comprises determining a scalar multiplication product of a predetermined scalar in the second selected basis and one of the first auxiliary points. ### Claim 11 (depends on 9) The cryptographic correspondent device of claim 9, wherein determining the linear combination comprises determining a scalar multiplication product of a predetermined scalar in the first selected basis and one of the first auxiliary points. ### Claim 12 (depends on 9) The cryptographic correspondent device of claim 9, wherein determining the linear combination comprises determining a scalar multiplication product of a predetermined scalar in the first selected basis, a predetermined scalar in the second selected basis, and one of the first auxiliary points. ### Claim 13 (depends on 12) The cryptographic correspondent device of claim 12, further comprising determining whether the values of the first public key are in the right domain by determining that such values are in defined torsion subgroups. ### Claim 14 (depends on 9) The cryptographic correspondent device of claim 9, wherein determining the linear combination comprises determining a sum of a first private scalar multiplied by the first auxiliary point and a second private scalar multiplied by the second auxiliary point. ### Claim 15 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. ### Claim 16 (depends on 15) The elliptic curve cryptographic scheme of claim 15, wherein the common key is static. ### Claim 17 (depends on 15) The elliptic curve cryptographic scheme of claim 15, wherein generating a common key further comprises determining a domain of each of the first-acquired auxiliary points and the second-acquired auxiliary points. ### Claim 18 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. ### Claim 19 (depends on 18) The cryptographic correspondent device of claim 18, wherein the common key is static. ### Claim 20 (depends on 18) The cryptographic correspondent device of claim 18, wherein generating a common key further comprises determining a domain of each of the first-acquired auxiliary points and the second-acquired auxiliary points.