Digital signature scheme
In cryptography, the Rabin signature algorithm is a method of digital signature originally proposed by Michael O. Rabin in 1978.[1][2][3]
The Rabin signature algorithm was one of the first digital signature schemes proposed. By introducing the use of hashing 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[4] and a security guarantee relative to the difficulty of integer factorization,[2][3][5] which has not been proven for RSA.
However, Rabin signatures have seen relatively little use or standardization outside IEEE P1363[6] 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
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 ![{\displaystyle x(x+b)\equiv H(m,u){\pmod {n}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- 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 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;[4] 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
.[3]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
.[5]
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[3] and resilience to collision attacks on fixed hash functions.[7][8][9]
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.[10][3][6][4]
The Rabin signature scheme was later tweaked by Williams in 1980[10] 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.[4][6]Further variants allow tradeoffs between signature size and verification speed, partial message recovery, signature compression (down to one-half size), and public key compression (down to one-third size), still without sacrificing security.[4]
Variants without the hash function have been published in textbooks,[11][12] 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,[2] 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,
, [the 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.[13]
References
- ^ Rabin, Michael O. (1978). "Digitalized Signatures". In DeMillo, Richard A.; Dobkin, David P.; Jones, Anita K.; Lipton, Richard J. (eds.). Foundations of Secure Computation. New York: Academic Press. pp. 155–168. ISBN 0-12-210350-5.
- ^ a b c Rabin, Michael O. (January 1979). Digitalized Signatures and Public Key Functions as Intractable as Factorization (PDF) (Technical report). Cambridge, MA, United States: MIT Laboratory for Computer Science. TR-212.
- ^ a b c d e Bellare, Mihir; Rogaway, Phillip (May 1996). Maurer, Ueli (ed.). The Exact Security of Digital Signatures—How to Sign with RSA and Rabin. Advances in Cryptology – EUROCRYPT ’96. Lecture Notes in Computer Science. Vol. 1070. Saragossa, Spain: Springer. pp. 399–416. doi:10.1007/3-540-68339-9_34. ISBN 978-3-540-61186-8.
- ^ a b c d e Bernstein, Daniel J. (January 31, 2008). RSA signatures and Rabin–Williams signatures: the state of the art (Report). (additional information at https://cr.yp.to/sigs.html)
- ^ a b Bernstein, Daniel J. (April 2008). Smart, Nigel (ed.). Proving tight security for Rabin–Williams signatures. Advances in Cryptology – EUROCRYPT 2008. Lecture Notes in Computer Science. Vol. 4965. Istanbul, Turkey: Springer. pp. 70–87. doi:10.1007/978-3-540-78967-3_5. ISBN 978-3-540-78966-6.
- ^ a b c IEEE Standard Specifications for Public-Key Cryptography. IEEE Std 1363-2000. Institute of Electrical and Electronics Engineers. August 25, 2000. doi:10.1109/IEEESTD.2000.92292. ISBN 0-7381-1956-3.
- ^ Bellare, Mihir; Rogaway, Phillip (August 1998). Submission to IEEE P1393—PSS: Provably Secure Encoding Method for Digital Signatures (PDF) (Report). Archived from the original (PDF) on 2004-07-13.
- ^ Halevi, Shai; Krawczyk, Hugo (August 2006). Dwork, Cynthia (ed.). Strengthening Digital Signatures via Randomized Hashing (PDF). Advances in Cryptology – CRYPTO 2006. Lecture Notes in Computer Science. Vol. 4117. Santa Barbara, CA, United States: Springer. pp. 41–59. doi:10.1007/11818175_3.
- ^ Dang, Quynh (February 2009). Randomized Hashing for Digital Signatures (Report). NIST Special Publication. Vol. 800–106. United States Department of Commerce, National Institute for Standards and Technology. doi:10.6028/NIST.SP.800-106.
- ^ a b Williams, Hugh C. "A modification of the RSA public-key encryption procedure". IEEE Transactions on Information Theory. 26 (6): 726–729. doi:10.1109/TIT.1980.1056264. ISSN 0018-9448.
- ^ Menezes, Alfred J.; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). "§11.3.4: The Rabin public-key signature scheme". Handbook of Applied Cryptography (PDF). CRC Press. pp. 438–442. ISBN 0-8493-8523-7.
- ^ Galbraith, Steven D. (2012). "§24.2: The textbook Rabin cryptosystem". Mathematics of Public Key Cryptography. Cambridge University Press. pp. 491–494. ISBN 978-1-10701392-6.
- ^ Elia, Michele; Schipani, David (2011). On the Rabin signature (PDF). Workshop on Computational Security. Centre de Recerca Matemàtica, Barcelona, Spain.
External links
- Rabin–Williams signatures at cr.yp.to