MATEMÁTICAS Y CRIPTOGRAFÍA

A nadie parece extrañar la frase “…hoy vivimos en la sociedad de la información…”, cada día cualquiera de nosotros almacena, procesa e incluso transmite información de forma digital. Sin duda, desde los años ochenta, el uso de los ordenadores, de las redes locales, de Internet, del correo electrónico, de la telefonía inalámbrica, del wireless… es parte de nuestra vida diaria.

La contrapartida del uso de toda esa información sobre redes abiertas son los problemas de privacidad y seguridad. Se deben pues buscar mecanismos tanto legales como científico-técnicos para garantizar ambas. Un ejemplo de plena actualidad es la puesta en marcha del Documento Nacional de Identidad electrónico.

La Criptografía es la parte científico-técnica que trata de garantizar los fundamentos de la seguridad en la transmisión de la información.
We live in an information-based society. The techniques to keep secret the information are the field of study of cryptography. The cryptography are the mathematical tools related to confidentiality, data integrity, entity authentication and data origin authentication.

There are two forms of encryption: Symmetric Key Encryption which is fast but both parties need to know a shared secret key and Public Key Encryption which is show but only one party needs to keep a key private.

The first and most popular public key exchange algorithm is RSA. The security of RSA is based on the intractability of the integer factorization problem. Diffie-Hellman key exchange relies on difficulty of computing discrete logarithms. There are a few other key exchange schemes that used in practice, for example, the Digital Signature Algorithms (DSA) and the Elliptic Curve Digital Signature (ECDSA). The security of those schemes is based on the discrete logarithm problem in the multiplicative group of points of an elliptic curve over finite field.

One of fundamental tools used in information security is the signature. From practical view point the main ingredient of signature in cryptology are the hash functions. The main role of crytographic hash function is in the provision of digital signatures. Since hash funtions are generally faster than digital signature algorithms, it is typical to compute the digital signature to some document by computing the signature on the document’s hash value, which is small compared to the document itself. It was a big surprise that vulnerability of standart hash funciones was announced in Feb. 2005. The attack primarily affects some digital signature applications, including timestamping and certificate signing operations, where one party prepares a message form the generation of digital signature by second party, and third parties then virefy the signature.

When quantum computers reach aproximately 30 to 40 q-bits they will start to have the speed (parallelism) needed to attack the methods society uses to Project data processes, including encryption, digital signatures, random number generators, key transmission, and other security algorithms. In particular all the standarts used nowdays will became obsolete and one should be prepared to have ready several other cryptosystems.

Second Cryptografy Hash Workshop: http://www.csrc.nist.gov/pki/HashWorkshop/index.html
Mathematics and Internet security: http://www.mathaware.org/mam/06/

Fuente: C. XIII, Matemáticas y Criptografía, de Matemáticas en la Frontera, Colección Madri+d.

Por Alejandro Melle Hernández
Universidad Complutense de Madrid
Departamento de Álgebra
Facultad de Matemáticas

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