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cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
, the Rabin signature algorithm is a method of digital signature originally proposed by
Michael O. Rabin Michael Oser Rabin (; born September 1, 1931) is an Israeli mathematician, computer scientist, and recipient of the Turing Award. Biography Early life and education Rabin was born in 1931 in Breslau, Germany (today Wrocław, in Poland), th ...
in 1978. 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 e=2, but this leads to qualitative differences that enable more efficient implementation (additional information at https://cr.yp.to/sigs.html) and a security guarantee relative to the difficulty of
integer factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...
, 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 (mathematics), function that can be used to map data (computing), data of arbitrary size to fixed-size values, though there are some hash functions that support variable-length output. The values returned by a ...
H(m, u) of a message m and k-bit randomization string u. ; Public key : A public key is a pair of integers (n, b) with 0 \leq b < n and n odd. b is chosen arbitrarily and may be a fixed constant. ; Signature : A signature on a message m is a pair (u, x) of a k-bit string u and an integer x such that x (x + b) \equiv H(m, u) \pmod n. ; Private key : The private key for a public key (n, b) is the secret odd prime factorization p\cdot q of n, chosen uniformly at random from some large space of primes. ; Signing a message : To make a signature on a message m using the private key, the signer starts by picking a k-bit string u uniformly at random, and computes c := H(m, u). Let d = (b/2) \bmod n. If c + d^2 is a
quadratic nonresidue In number theory, an integer ''q'' is a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; that is, if there exists an integer ''x'' such that :x^2\equiv q \pmod. Otherwise, ''q'' is a quadratic nonresidue modu ...
modulo n, the signer starts over with an independent random u. Otherwise, the signer computes \begin x_p &:= \Bigl(-d \pm \sqrt\Bigr) \bmod p, \\ x_q &:= \Bigl(-d \pm \sqrt\Bigr) \bmod q, \end using a standard algorithm for computing square roots modulo a prime—picking p \equiv q \equiv 3 \pmod 4 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 c. x_p and x_q satisfy the equations \begin x_p (x_p + b) &\equiv H(m,u) \pmod p, \\ x_q (x_q + b) &\equiv H(m,u) \pmod q. \end The signer then uses the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
to solve the system \begin x &\equiv x_p \pmod p, \\ x &\equiv x_q \pmod q, \end for x, so that x satisfies x (x + b) \equiv H(m,u) \pmod n as required. The signer reveals (u, x) as a signature on m. : The number of trials for u before x (x + b) \equiv H(m,u) \pmod n can be solved for x is geometrically distributed with an average around 4 trials, because about 1/4 of all integers are quadratic residues modulo n.


Security

Security against any adversary defined generically in terms of a hash function H (i.e., security in the
random oracle model In cryptography, a random oracle is an oracle (a theoretical black box) that responds to every ''unique query'' with a (truly) random response chosen uniformly from its output domain. If a query is repeated, it responds the same way every tim ...
) follows from the difficulty of factoring n: Any such adversary with high probability of success at forgery can, with nearly as high probability, find two distinct square roots x_1 and x_2 of a random integer c modulo n. If x_1 \pm x_2 \not\equiv 0 \pmod n then \gcd(x_1 \pm x_2, n) is a nontrivial factor of n, since ^2 \equiv ^2 \equiv c \pmod n so n \mid ^2 - ^2 = (x_1 + x_2) (x_1 - x_2) but n \nmid x_1 \pm x_2. Formalizing the security in modern terms requires filling in some additional details, such as the codomain of H; if we set a standard size K for the prime factors, 2^ < p < q < 2^K, then we might specify H\colon \^* \times \^k \to \^K. 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


Removing b

The quantity b in the public key adds no security, since any algorithm to solve congruences x (x + b) \equiv c \pmod n for x given b and c can be trivially used as a subroutine in an algorithm to compute square roots modulo n and vice versa, so implementations can safely set b = 0 for simplicity; b was discarded altogether in treatments after the initial proposal. After removing b, the equations for x_p and x_q in the signing algorithm become: \begin x_p &:= \pm \sqrt \bmod p, \\ x_q &:= \pm \sqrt \bmod q. \end


Rabin-Williams

The Rabin signature scheme was later tweaked by Williams in 1980 to choose p \equiv 3 \pmod 8 and q \equiv 7 \pmod 8, and replace a square root x by a tweaked square root (e, f, x), with e = \pm1 and f \in \, so that a signature instead satisfies e f x^2 \equiv H(m, u) \pmod n, which allows the signer to create a signature in a single trial without sacrificing security. This variant is known as Rabin–Williams.


Others

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. 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 x = 2 can be forged by anyone as a valid signature on the message m = 4 if the signature verification equation is x^2 \equiv m \pmod n instead of x^2 \equiv H(m, u) \pmod n. In the original paper, the hash function H(m, u) was written with the notation C(MU), with ''C'' for ''compression'', and using juxtaposition to denote concatenation of M and U as bit strings:
By convention, when wishing to sign a given message, M, he signerP adds as suffix a word U of an agreed upon length k. The choice of U is randomized each time a message is to be signed. The signer now compresses M_1 = MU by a hashing function to a word C(M_1) = c, so that as a binary number c \leq n
This notation has led to some confusion among some authors later who ignored the C part and misunderstood MU 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