Rabin signature algorithm

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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.


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.
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 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 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 , and 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]


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, and public key compression.[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]


  1. ^ Rabin, Michael O. (1978). "Digitalized Signatures". In DeMillo, Richard A.; Dobkin, David P.; Jones, Anita K.; Lipton, Richard J. (eds.). Foundations of Secure Computation. 111 Fifth Avenue, New York, NY 10003, United States: Academic Press. pp. 155–168. ISBN 0-12-210350-5.CS1 maint: location (link)
  2. ^ 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.
  3. ^ 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. 1070. Saragossa, Spain: Springer. pp. 399–416. doi:10.1007/3-540-68339-9_34. ISBN 978-3-540-61186-8.
  4. ^ a b c d e Bernstein, Daniel J. (January 31, 2008). RSA signatures and Rabin–Williams signatures: the state of the art (Report).
  5. ^ 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. 4965. Istanbul, Turkey: Springer. pp. 70–87. doi:10.1007/978-3-540-78967-3_5. ISBN 978-3-540-78966-6.
  6. ^ a b c IEEE Std 1363-2000: IEEE Standard Specifications for Public-Key Cryptography. Institute of Electrical and Electronics Engineers. August 25, 2000. doi:10.1109/IEEESTD.2000.92292. ISBN 0-7381-1956-3.
  7. ^ 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.
  8. ^ 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. 4117. Santa Barbara, CA, United States: Springer. pp. 41–59. doi:10.1007/11818175_3.
  9. ^ Dang, Quynh (February 2009). Randomized Hashing for Digital Signatures (Report). NIST Special Publication. 800–106. United States Department of Commerce, National Institute for Standards and Technology. doi:10.6028/NIST.SP.800-106.
  10. ^ 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.
  11. ^ 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.
  12. ^ 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.
  13. ^ Elia, Michele; Schipani, David (2011). On the Rabin signature (PDF). Workshop on Computational Security. Centre de Recerca Matemàtica, Barcelona, Spain.

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