Power analysis attack countermeasure for the ECDSA
Execution of the Elliptic Curve Digital Signature Algorithm (ECDSA) requires determination of a signature, which determination involves arithmetic operations. Some of the arithmetic operations employ a long term cryptographic key. It is the execution of these arithmetic operations that can make the execution of the ECDSA vulnerable to a power analysis attack. In particular, an attacker using a power analysis attack may determine the long term cryptographic key. By modifying the sequence of operations involved in the determination of the signature and the inputs to those operations, power analysis attacks may no longer be applied to determine the long term cryptographic key.
1. A method of publishing a signature related to a message in a manner that counters power analysis attacks, wherein a private cryptographic key and a base point on a given elliptic curve have been selected, said base point having a prime order, said method comprising:
receiving, by a processing device, said message;
obtaining, by the processing device, a hash of said message;
selecting, by the processing device, a random integer;
obtaining, by the processing device, a non-zero first element of said signature based on said base point and said random integer;
obtaining, by the processing device, a non-zero second element of said signature based on said random integer, said hash, said first element, said private cryptographic key and a modular inverse of said random integer, wherein said obtaining said second element involves:
determining a first modular multiplication product of said private cryptographic key and said modular inverse of said random integer; and
determining a second modular multiplication product of said first modular multiplication product and a modular multiple of said first element; and
publishing, by the processing device, said first element of said signature and said second element of said signature.
2. The method of claim 1 , wherein said modular multiple of said first element is said first element.
3. The method of claim 2 wherein said obtaining said second element evaluating:
s =[k −1 m+(k −1 d A ) r]mod g
wherein
s is said second element;
k is said random integer;
m is said hash;
d A is said private cryptographic key;
r is said first element; and
g is said prime order of said base point.
4. A mobile communication device for publishing a signature related to a message in a manner that counters power analysis attacks, wherein a private cryptographic key and a base point on a given elliptic curve have been selected, said base point having a prime order, said apparatus comprising:
a processor configured to:
receive said message;
obtain a hash of said message;
select a random integer;
obtain a non-zero first element of said signature based on said base point and said random integer;
obtain a non-zero second element of said signature based on said random integer, said hash, said first element, said private cryptographic key and a modular inverse of said random integer, by:
determining a first modular multiplication product of said private cryptographic key and said modular inverse of said random integer; and
determining a second modular multiplication product of said first modular multiplication product and a modular multiple of said first element; and
publish said first element of said signature and said second element of said signature.
5. A non-transitory computer readable medium containing computer-executable instructions that, when executed on a processor given a private cryptographic key and a base point on a given elliptic curve, said base point having a prime order, cause said processor to:
receive a message;
obtain a hash of said message;
select a random integer;
obtain a non-zero first element of a signature based on said base point and said random integer;
obtain a non-zero second element of said signature based on said random integer, said hash, said first element, said private cryptographic key and a modular inverse of said random integer, by:
determining a first modular multiplication product of said private cryptographic key and said modular inverse of said random integer; and
determining a second modular multiplication product of said first modular multiplication product and a modular multiple of said first element; and
publish said first element of said signature and said second element of said signature.