Methods and systems for enhanced data-centric scalar multiplicative homomorphic encryption systems using geometric algebra

A method for encrypting a numeric message data value (M) on a source computing device in order to transfer a cryptotext multivector (C) encrypted representation of said numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of said cryptotext multivector (C) and an unencrypted scalar data value (V) and deliver a result of said homomorphic scalar multiplication to a destination computing device, the method comprising:

• distributing by said source computing device said numeric message data value (M) into coefficients of a message multivector (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device;
• distributing by said source computing device a shared secret numeric value (SS) into coefficients of a shared secret multivector (SS) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (SS) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
• encrypting by said source computing device said cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M) and said shared secret multivector (SS);
• and
• sending by said source computing device said cryptotext multivector (C) to said intermediary computing system.

A method for performing homomorphic scalar multiplication on an intermediary computer system of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the method comprising:

• receiving by said intermediary computing system said cryptotext multivector (C) sent by said source computing device;
• multiplying by said intermediary computing system said unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC);
• and
• sending by said intermediary computing system said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device.

A method for decrypting a scalar multiplicative result cryptotext multivector (SMRC) on a destination computing device received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the method comprising:

• receiving by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
• distributing by said source computing device a shared secret numeric value (SS) into a shared secret multivector (SS) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (SS) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
• decrypting by said destination computing device said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse (SS−1) of said shared secret multivector (SS) into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for an encryption process of said cryptotext multivector (C);
• and
• converting by said destination computing device said scalar multiplicative result multivector (SMR) into a scalar multiplicative result data value (SMR) in accord with said homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device such that said scalar multiplicative result value (SMR) is equal to a multiplication product of an unencrypted numeric message data value (M) represented by said cryptotext multivector (C) and said unencrypted scalar data value (V).

A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system for scalar multiplicative homomorphic multiplication of a cryptotext encrypted data representation of a corresponding plain text data value and an unencrypted scalar data value, the scalar multiplicative homomorphic EDCE system comprising:

• a source computing device, wherein said source computing device further comprises:
• a source numeric message distribution subsystem that distributes a numeric message data value (M) into coefficients of a message multivector (M) representing said numeric message data value (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and a destination computing device;
• a source numeric shared secret distribution subsystem that distributes a shared secret numeric value (SS) into coefficients of a shared secret multivector (SS) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, such that said shared secret numeric value (SS) is known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including an intermediary computing system;
• a source encryption subsystem that encrypts a cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M) and said shared secret multivector (SS);
• and
• a source send subsystem that sends said cryptotext multivector (C) to said intermediary computing system;
• said intermediary computing system, wherein said intermediary computing system further comprises:
• an intermediary receive subsystem that receives said cryptotext multivector (C) sent by said source computing device;
• an intermediary homomorphic scalar multiplication subsystem that multiplies an unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC);
• and
• an intermediary send subsystem that sends said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device;
• and
• said destination computing device, wherein said destination computing device further comprises:
• a destination receive subsystem that receives said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
• a destination numeric shared secret distribution subsystem that distributes said shared secret numeric value (SS) into said shared secret multivector (SS) in accord with said shared secret coefficient distribution algorithm;
• a destination decryption subsystem that decrypts said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse (SS−1) of said shared secret multivector (SS) into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for said encryption process of said cryptotext multivector (C);
• and
• a destination convert multivector subsystem that converts said scalar multiplicative result multivector (SMR) into a scalar multiplicative result value (SMR) in accord with said homomorphic preserving mathematical relationship such that said scalar multiplicative result value (SMR) is equal to a multiplication product of said unencrypted numeric message data value (M) and said unencrypted scalar data value (V).

A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system source computing device for encrypting a numeric message data value (M) in order to transfer a cryptotext multivector (C) encrypted representation of said numeric message data value (M) to an intermediary computing system that will perform homomorphic scalar multiplication of said cryptotext multivector (C) and an unencrypted scalar data value (V) and deliver a result of said homomorphic scalar multiplication to a destination computing device, the scalar multiplicative homomorphic EDCE system source computing device comprising:

• a source numeric message distribution subsystem that distributes said numeric message data value (M) into coefficients of a message multivector (M) in accord with a homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device;
• a source numeric shared secret distribution subsystem that distributes a shared secret numeric value (SS) into coefficients of a shared secret multivector (SS) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (SS) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
• a source encryption subsystem that encrypts said cryptotext multivector (C) as an encryption function of at least one Geometric Algebra geometric product operation on said message multivector (M) and said shared secret multivector (SS);
• and
• a source send subsystem that sends said cryptotext multivector (C) to said intermediary computing system.

A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system intermediary computing system for performing homomorphic scalar multiplication of a cryptotext multivector (C) encrypted data representation of a corresponding plain text numeric data value received from a source computing device and an unencrypted scalar data value (V) and delivering a homomorphic scalar multiplicative result cryptotext multivector (SMRC) to a destination computing device, the scalar multiplicative homomorphic EDCE system intermediary computing system comprising:

• an intermediary receive subsystem that receives said cryptotext multivector (C) sent by said source computing device;
• an intermediary homomorphic scalar multiplication subsystem that multiplies said unencrypted scalar data value (V) and said cryptotext multivector (C) using scalar-vector multiplication in order to obtain a scalar multiplicative result cryptotext multivector (SMRC);
• and
• an intermediary send subsystem that sends said scalar multiplicative result cryptotext multivector (SMRC) to said destination computing device.

A scalar multiplicative homomorphic Enhanced Data-Centric Encryption (EDCE) system destination computing device for decrypting a scalar multiplicative result cryptotext multivector (SMRC) received from an intermediary computing system that performed homomorphic scalar multiplication of a cryptotext multivector (C) originated from a source computing device and an unencrypted scalar data value (V), the scalar multiplicative homomorphic EDCE system destination computing device comprising:

• a destination receive subsystem that receives said scalar multiplicative result cryptotext multivector (SMRC) sent by said intermediary computing system;
• a destination numeric shared secret distribution subsystem that distributes a shared secret numeric value (SS) into a shared secret multivector (SS) in accord with a shared secret coefficient distribution algorithm that is known to said source computing device and said destination computing device, said shared secret numeric value (SS) being known or knowable to said source computing device and said destination computing device, but is kept secret from other devices not intended to have access to said numeric message data including said intermediary computing system;
• a destination decryption subsystem that decrypts said scalar multiplicative result cryptotext multivector (SMRC) as a decryption function of at least one Geometric Algebra geometric product operation on said scalar multiplicative result cryptotext multivector (SMRC) and an inverse (SS−1) of said shared secret multivector (SS) into a scalar multiplicative result multivector (SMR) such that said decryption function provides a corresponding decryption operation for an encryption process of said cryptotext multivector (C);
• and
• a destination convert multivector subsystem that converts said scalar multiplicative result multivector (SMR) into a scalar multiplicative result data value (SMR) in accord with said homomorphic preserving mathematical relationship between an unencrypted numeric data value and multivector coefficients representing said unencrypted numeric data value that is known to said source computing device and said destination computing device such that said scalar multiplicative result value (SMR) is equal to a multiplication product of an unencrypted numeric message data value (M) represented by said cryptotext multivector (C) and said unencrypted scalar data value (V).