In cryptography, concrete security or exact security is a practice-oriented approach that aims to give more precise estimates of the computational complexities of adversarial tasks than polynomial equivalence would allow. The German Lorenz cipher machine Cryptography or cryptology is a field of mathematics and computer science concerned with information security and related issues, particularly encryption and authentication. ... In cryptography, an adversary (rarely opponent, enemy) is a malicious entity whose aim is to prevent the users of the cryptosystem from achieving their goal (primarily privacy, integrity and availability of data). ... In computational complexity theory a polynomial-time reduction is a reduction which is computable by a deterministic Turing machine in polynomial time. ...

Traditionally, provable security is asymptotic: it classifies the hardness of computational problems using polynomial-time reducibility. Secure schemes are defined to be those in which the advantage of any computationally bounded adversary is negligible . While such a theoretical guarantee is important, in practice one needs to know exactly how efficient a reduction is because of the need to instantiate the security parameter - it is not enough to know that "sufficiently large" security parameters will do. An inefficient reduction results either in the success probability for the adversary or the resource requirement of the scheme being greater than desired. In cryptography, a system is said to have provable security if its security requirements are stated formally in an adversarial model, as opposed to heuristically, and there is a proof (called a reduction) that these security requirements can be met provided that some well studied cryptographic primitive (such as RSA... In cryptography, an adversary (rarely opponent, enemy) is a malicious entity whose aim is to prevent the users of the cryptosystem from achieving their goal (primarily privacy, integrity and availability of data). ... In cryptography, a negligible function is one that approaches zero faster than the reciprocal of any polynomial. ...

Concrete security parametrizes all the resources available to the adversary, such as running time and memory, and other resources specific to the system in question, such as the number of plaintexts it can obtain or the number of queries it can make to any oracles available. Then the advantage of the adversary is upper bounded as a function of these resources and of the problem size. It is often possible to give a lower bound (i.e, an adversarial strategy) matching the upper bound, hence the name exact security.

References

• M. Bellare, A. Desai, E. Jokipii and P. Rogaway. A Concrete Security Treatment of Symmetric Encryption: Analysis of the DES Modes of Operation.
• M. Bellare and P. Rogaway. The Exact Security of Digital Signatures: How to Sign with RSA and Rabin