(The real-world public-key algorithm
involves much larger numbers.) He does some math of his own in advance to discover the corresponding private key for that public key, which in this case, is 33 and 3.
"Symmetric-key cryptography works well because encrypting and decrypting operations can take place about 1,000 times faster than when you use a public-key algorithm
," explained Freescale's Bettelheim.
The LSB of each pixel in a block is stripped off, and the remaining MSBs of the pixels and the image size parameters are hashed and the result encrypted using a public-key algorithm
. The resulting cipher text and the binary watermark image are combined using an exclusive-OR function; the result is then embedded into the LSB of the block.
The best advice on encryption is to use a public-key algorithm
available to hackers, that has been attacked and tested by many people, but not yet cracked.
Each encrypted message key copy is placed in a Key-Info field, following an identifier for the public-key algorithm
used to encrypt the copy of the message key.
A number of works based on hashing and message digests (MDs) were then proposed for achieving authentication and integrity with reduced overhead as a tradeoff for removing secrecy measures during transmission in order to achieve improved performance when applied in scenarios involving public-key algorithms
The introduction of elliptic curve cryptography by Neal Koblitz  and Victor Miller  independently and simultaneously in the mid-1980s has yielded new public-key algorithms
based on the discrete logarithm problem.
Also, because ECC offers more security per bit than other public-key algorithms
, it doesn't compromise performance, a feature especially important in constrained environments such as mobile devices and chips.
are orders of magnitude slower in execution than symmetric key algorithms.
Over the last 20 years several strong cryptographic algorithms(2) have emerged, including the Data Encryption Standard (DES), and the public-key algorithms
, Diffie-Hellman and RSA.