In
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
, the Rabin signature algorithm is a method of
digital signature originally proposed by
Michael O. Rabin
Michael Oser Rabin ( he, מִיכָאֵל עוזר רַבִּין; born September 1, 1931) is an Israeli mathematician and computer scientist and a recipient of the Turing Award.
Biography Early life and education
Rabin was born in 1931 in ...
in 1978.
The Rabin signature algorithm was one of the first digital signature schemes proposed. By introducing the use of
hashing
Hash, hashes, hash mark, or hashing may refer to:
Substances
* Hash (food), a coarse mixture of ingredients
* Hash, a nickname for hashish, a cannabis product
Hash mark
* Hash mark (sports), a marking on hockey rinks and gridiron football fiel ...
as an essential step in signing, it was the first design to meet what is now the modern standard of security against forgery,
existential unforgeability under chosen-message attack, assuming suitably scaled parameters.
Rabin signatures resemble
RSA signatures with 'exponent
', but this leads to qualitative differences that enable more efficient implementation
and a security guarantee relative to the difficulty of
integer factorization,
which
has not been proven for RSA.
However, Rabin signatures have seen relatively little use or standardization outside
IEEE P1363
IEEE P1363 is an Institute of Electrical and Electronics Engineers (IEEE) standardization project for public-key cryptography. It includes specifications for:
* Traditional public-key cryptography (IEEE Std 1363-2000 and 1363a-2004)
* Lattice-ba ...
in comparison to RSA signature schemes such as
RSASSA-PKCS1-v1_5 and
RSASSA-PSS.
Definition
The Rabin signature scheme is parametrized by a randomized
hash function
A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called ''hash values'', ''hash codes'', ''digests'', or simply ''hashes''. The values are usually ...
of a message
and
-bit randomization string
.
; Public key
: A public key is a pair of integers
with
and
odd.
; Signature
: A signature on a message
is a pair
of a
-bit string
and an integer
such that
; Private key
: The private key for a public key
is the secret odd prime factorization
of
, chosen uniformly at random from some space of large primes. Let
,
, and
. To make a signature on a message
, the signer picks a
-bit string
uniformly at random, and computes
. If
is a
quadratic nonresidue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic no ...
modulo
, then the signer throws away
and tries again. Otherwise, the signer computes
using a
standard algorithm for computing square roots modulo a prime—picking
makes it easiest. Square roots are not unique, and different variants of the signature scheme make different choices of square root;
in any case, the signer must ensure not to reveal two different roots for the same hash
. The signer then uses the
Chinese remainder theorem to solve the system
for
. The signer finally reveals
.
Correctness of the signing procedure follows by evaluating
modulo
and
with
as constructed. For example, in the simple case where
,
is simply a square root of
modulo
. The number of trials for
is geometrically distributed with expectation around 4, because about 1/4 of all integers are quadratic residues modulo
.
Security
Security against any adversary defined generically in terms of a hash function
(i.e., security in the
random oracle model) follows from the difficulty of factoring
:
Any such adversary with high probability of success at forgery can, with nearly as high probability, find two distinct square roots
and
of a random integer
modulo
.
If
then
is a nontrivial factor of
, since
so
but
.
Formalizing the security in modern terms requires filling in some additional details, such as the codomain of
; if we set a standard size
for the prime factors,
, then we might specify
.
Randomization of the hash function was introduced to allow the signer to find a quadratic residue, but randomized hashing for signatures later became relevant in its own right for tighter security theorems
and resilience to collision attacks on fixed hash functions.
Variants
The quantity
in the public key adds no security, since any algorithm to solve congruences
for
given
and
can be trivially used as a subroutine in an algorithm to compute square roots modulo
and vice versa, so implementations can safely set
for simplicity;
was discarded altogether in treatments after the initial proposal.
The Rabin signature scheme was later tweaked by
Williams in 1980
to choose
and
, and replace a square root
by a tweaked square root
, with
and
, so that a signature instead satisfies
which allows the signer to create a signature in a single trial without sacrificing security.
This variant is known as Rabin–Williams.
Further variants allow tradeoffs between signature size and verification speed, partial message recovery, signature compression, and public key compression.
Variants without the hash function have been published in textbooks,
crediting Rabin for exponent 2 but not for the use of a hash function.
These variants are trivially broken—for example, the signature
can be forged by anyone as a valid signature on the message
if the signature verification equation is
instead of
.
In the original paper,
the hash function
was written with the notation
, with ''C'' for ''compression'', and using juxtaposition to denote concatenation of
and
as bit strings:
By convention, when wishing to sign a given message, , he signer adds as suffix a word of an agreed upon length .
The choice of is randomized each time a message is to be signed.
The signer now compresses by a hashing function to a word , so that as a binary number …
This notation has led to some confusion among some authors later who ignored the
part and misunderstood
to mean multiplication, giving the misapprehension of a trivially broken signature scheme.
References
{{reflist
External links
Rabin–Williams signatures at cr.yp.to
Digital signature schemes