Elliptic curve arithmetic in script

A computer-implemented method for performing elliptic curve arithmetic using blockchain transactions, wherein the method is performed by a script engine and comprises:

• obtaining a first blockchain transaction, wherein the first blockchain transaction comprises a first script, wherein the first script comprises a modular inversion script, and wherein the modular inversion script is configured to, when executed, by a script engine:
• obtain a target integer (x) and a modulus value (p), wherein the target integer and the modulus value are co-prime, and the modulus value is a prime number, and
• output a modular inverse (y) of the target integer, wherein the modular inverse is generated by:
• generating a sequence of squaring values, wherein a first squaring value is equal to the target integer (x) and wherein each subsequent squaring value of the sequence of squaring values is generated by multiplying an immediately previous squaring value by itself;
• for each set bit of a reduced modulus value (p−2), wherein the reduced modulus value is an integer of value two less than the modulus value and is expressed in binary, including a corresponding squaring value of the sequence of squaring values in a result;
• multiplying together the squaring values of the result to generate a resultant value;
• and
• computing a modulus of the resultant value with respect to the modulus value (p), wherein the modulus of the resultant value is output as the modular inverse (y), and
• obtaining a second blockchain transaction, wherein the second blockchain transaction comprises a second script, wherein the second script comprises the target integer (x) and the modulus value (p);
• and
• executing the first script together with the second script to generate the modular inverse of the target integer.

Computer equipment comprising:

• memory comprising one or more memory units;
• and
• processing apparatus comprising one or more processing units, wherein the memory stores code arranged to run on the processing apparatus, the code being configured so as when run on the processing apparatus, the processing apparatus performs a method for enabling elliptic curve arithmetic to be performed using blockchain transactions, wherein the method is performed by a script generator and comprises:
• generating a first script of a first blockchain transaction, wherein the first script comprises a modular inversion script, and wherein the modular inversion script is configured to, when executed, by a script engine:
• obtain a target integer (x) and a modulus value (p), wherein the target integer and the modulus value are co-prime and the modulus value is a prime number, and
• output a modular inverse (y) of the target integer, wherein the modular inverse is generated by:
• generating a sequence of squaring values, wherein a first squaring value is equal to the target integer (x) and wherein each subsequent squaring value of the sequence of squaring values is generated by multiplying an immediately previous squaring value by itself;
• for each set bit of a reduced modulus value (p−2), wherein the reduced modulus value is an integer of value two less than the modulus value and is expressed in binary, including a corresponding squaring value of the sequence of squaring values in a result;
• multiplying together the squaring values of the result to generate a resultant value;
• and
• computing a modulus of the resultant value with respect to the modulus value (p), wherein the modulus of the resultant value is output as the modular inverse (y), and
• outputting the first script for inclusion in the first blockchain transaction.

A computer program embodied on non-transitory computer-readable storage media and configured so as, when run on one or more processors, the one or more processors perform a method for enabling elliptic curve arithmetic to be performed using blockchain transactions, wherein the method is performed by a script generator and comprises:

• generating a first script of a first blockchain transaction, wherein the first script comprises a modular inversion script, and wherein the modular inversion script is configured to, when executed, by a script engine:
• obtain a target integer (x) and a modulus value (p), wherein the target integer and the modulus value are co-prime and the modulus value is a prime number, and
• output a modular inverse (y) of the target integer, wherein the modular inverse is generated by:
• generating a sequence of squaring values, wherein a first squaring value is equal to the target integer (x) and wherein each subsequent squaring value of the sequence of squaring values is generated by multiplying an immediately previous squaring value by itself;
• for each set bit of a reduced modulus value (p−2), wherein the reduced modulus value is an integer of value two less than the modulus value and is expressed in binary, including a corresponding squaring value of the sequence of squaring values in a result;
• multiplying together the squaring values of the result to generate a resultant value;
• and
• computing a modulus of the resultant value with respect to the modulus value (p), wherein the modulus of the resultant value is output as the modular inverse (v), and
• outputting the first script for inclusion in the first blockchain transaction.