Cryptography Reference
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may be secure if the message is “randomized” before RSA (or the other
schemes) is applied.
Secure message authentication schemes can be constructed using
pseudorandom functions (68). Specifically, the key-generation algo-
rithm consists of selecting a seed s
0 , 1
for such a function,
n , and the (only valid) tag of message x with
respect to the key s is f s ( x ). As in the case of our private-key encryp-
tion scheme, the proof of security of the current message authentication
scheme consists of two steps:
} →{
denoted f s :
0 , 1
0 , 1
(1) Prove that an idealized version of the scheme, in which one
uses a uniformly selected function F :
n ,rather
than the pseudorandom function f s , is secure (i.e., unforge-
(2) Conclude that the real scheme (as presented above) is secure
(because, otherwise one could distinguish a pseudorandom
function from a truly random one).
} →{
0 , 1
0 , 1
Note that the aforementioned message authentication scheme makes
an “extensive use of pseudorandom functions” (i.e., the pseudorandom
function is applied directly to the message, which requires a general-
ized notion of pseudorandom functions (cf. Section 3.3)). More ecient
schemes may be obtained either based on a more restricted use of a
pseudorandom function (cf., e.g., (17)) or based on other cryptographic
primitives (cf., e.g., (93)).
Constructing secure signature schemes seems more dicult than
constructing message authentication schemes. Nevertheless, secure sig-
nature schemes can be constructed based on any one-way function.
Theorem 6.2. ((103; 114), see (67, Sec. 6.4)): The following three
conditions are equivalent.
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