Arithmetic device and method

A system, comprising:

• a memory circuit having a storage region storing firmware;
• and
• a signature verification circuit configured to obtain a digital signature and having an arithmetic device outputting an arithmetic result on a finite field with characteristic P, the arithmetic device comprising:
• a hardware processor configured to
• perform readout processing of an input value,
• calculate an approximate value of a remainder resulting from dividing the input value by the characteristic P, the approximate value being calculated on a basis of a higher-order word of the input value and a higher-order word of the characteristic P,
• perform, for each word, arithmetic operations with respect to the input value by using a value being based on the approximate value and the characteristic P,
• output a first output value resulting from computing a value being based on the input value, the approximate value, and the characteristic P, and
• output a second output value resulting from comparing the first output value and the characteristic P,
• wherein the signature verification circuit is configured to:
• perform an arithmetic operation using the hardware processor on a first part of the signature using the arithmetic device,
• obtain a value using a public key and the first part of the signature,
• determine whether the value matches a second part of the signature different from the first part of the signature,
• when the value matches the second part of the signature, permit execution of the firmware to begin, and
• when the value does not match the second part of the signature, not permit execution of the firmware to begin.

A system, comprising:

• a memory circuit having a storage region storing firmware and a digital signature;
• and
• a signature verification circuit configured to obtain the digital signature having an arithmetic device outputting an arithmetic result on a finite field with characteristic P, the arithmetic device comprising:
• a hardware processor configured to
• perform readout processing of an input value,
• calculate an update matrix being based on the input value and the characteristic P,
• update, on a basis of the update matrix, the input value and the characteristic P,
• output a first output value resulting from computing a value being based on the input value, the update matrix, and the characteristic P, and
• output a second output value resulting from comparing the first output value and the characteristic P,
• wherein the signature verification circuit is configured to:
• perform an arithmetic operation using the hardware processor on a first part of the signature using the arithmetic device,
• obtain a value using a public key and the first part of the signature,
• determine whether the value matches a second part of the signature different from the first part of the signature,
• when the value matches the second part of the signature, permit execution of the firmware to begin, and
• when the value does not match the second part of the signature, not permit execution of the firmware to begin.

A signature verification method, comprising:

• storing firmware in a memory;
• obtaining a digital signature;
• performing an arithmetic operation using an arithmetic device on a first part of the signature;
• obtaining a value using a public key and the first part of the signature, and
• determining whether the value matches a second part of the signature different from the first part of the signature,
• wherein performing the arithmetic operation comprises controlling the arithmetic device to output an arithmetic result on a finite field with characteristic P using a hardware processor, comprising:
• performing readout processing of a plurality of input values;
• performing, for each word, arithmetic operations with respect to the plurality of input values by using a value being based on the characteristic P and a comparison value between each input value of the plurality of input values and the characteristic P;
• outputting a first output value resulting from computing a value being based on each input value of the plurality of input values, the comparison value, and the characteristic P;
• and
• outputting a second output value resulting from comparing the first output value and the characteristic P,
• when the value matches the second part of the signature, execution of the firmware is permitted to begin, and
• when the value does not match the second part of the signature, execution of the firmware does not begin.

A method of controlling an arithmetic device in a signature verification circuit of a system having a memory storing firmware, the arithmetic device outputting an arithmetic result on a finite field with characteristic P using a hardware processor, the method comprising:

• performing readout processing of an input value;
• calculating an approximate value of a remainder resulting from dividing the input value by the characteristic P, the approximate value being calculated on a basis of a higher-order word of the input value and a higher-order word of the characteristic P;
• performing, for each word, arithmetic operations with respect to the input value by using a value being based on the approximate value and the characteristic P;
• outputting a first output value resulting from computing a value being based on the input value, the approximate value, and the characteristic P;
• outputting a second output value resulting from comparing the first output value and the characteristic P;
• obtaining a digital signature;
• performing an arithmetic operation using the arithmetic device on a first part of the signature;
• obtaining a value using a public key and the first part of the signature using the signature verification circuit, and
• determining whether the value matches a second part of the signature different from the first part of the signature using the signature verification circuit,
• when the value matches the second part of the signature, execution of the firmware is permitted to begin, and
• when the value does not match the second part of the signature, execution of the firmware does not begin.

A method of controlling an arithmetic device in a signature verification circuit of a system having a memory storing firmware and a digital signature, the arithmetic device outputting an arithmetic result on a finite field with characteristic P using a hardware processor, the method comprising:

• performing readout processing of an input value;
• calculating an update matrix being based on the input value and the characteristic P;
• updating, on a basis of the update matrix, the input value and the characteristic P;
• outputting a first output value resulting from computing a value being based on the input value, the update matrix, and the characteristic P;
• outputting a second output value resulting from comparing the first output value and the characteristic P;
• obtaining the digital signature;
• performing an arithmetic operation using the arithmetic device on a first part of the signature;
• obtaining a value using a public key and the first part of the signature using the signature verification circuit, and
• determining whether the value matches a second part of the signature different from the first part of the signature using the signature verification circuit,
• when the value matches the second part of the signature, execution of the firmware is permitted to begin, and
• when the value does not match the second part of the signature, execution of the firmware does not begin.