Network arrangement for secure use of a private key remotely accessed through an open network

A first computing device including:

• a processor;
• and
• a non-transitory computer-readable medium including computer-executable instructions, that, when executed by the processor, cause the first computing device to carry out a method over a network infrastructure, for participating in completing cryptographically secured transactions with a server, using a first public/private key pair, wherein a first public key of the first public/private key pair is used by the server and the first private key of the first public/private key pair is currently stored on a second computing device that is accessible to the first computing device through an open network, and wherein the method comprises:
• providing a simultaneous user access to both the first computing device and the second computing device;
• establishing, during the providing the simultaneous user access, a connection over the open network between the first computing device and the second computing device by performing the following:
• generating, by the first computing device, a second public/private key pair, wherein a second public key of the second public/private key pair is provided to the second computing device;
• generating, by the first computing device, a message digest of a second public key of the second public/private key pair;
• presenting, by the first computing device via a display interface, a code value corresponding to the message digest to the user;
• transferring, using an action performed by the user, the code value corresponding to the message digest from the first computing device to the second computing device;
• and
• using, by the second computing device, the message digest to authenticate the second public key provided to the second computing device in association with the generating a second public/private key pair;
• sending, by the first computing device to the second computing device via the connection over the open network, a message requiring cryptographic processing using the first private key, wherein the message causes the second computing device to:
• perform a cryptographic operation on contents of the message using the first private key to render a cryptographically processed message;
• and
• send the cryptographically processed message to the first computing device via the connection over the open network;
• and
• receiving the cryptographically processed message over the open network from the second computing device;
• and
• providing the cryptographically processed message to the server.