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Encyclopedia of Cryptography and Security pp 532–536Cite as

RSA Problem

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Introduction

In RSA public-key encryption [30], Alice encrypts a plaintext M for Bob using Bob's public key (n, e) by computing the ciphertext

$$ C = M^{ e}\!\!\!\! \pmod{n}, $$
((1))

where n, the modulus, is the product of two or more large primes, and e, the public exponent, is an (odd) integer \(e \ge 3\) that is relatively prime to \(\phi(n)\), the order of the multiplicative group \({\bf Z} _n^\ast\). (See also Euler's totient function, modular arithmetic for background on these concepts.) Bob, who knows the corresponding RSA private key \((n,d)\), can easily decrypt, since \(de=1\pmod{\phi(n)}\) implies that

$$ M = C^d\!\!\!\! \pmod{n}. $$
((2))

An adversary may learn C by eavesdropping, and may very well also know Bob's public key; nonetheless such an adversary should not be able to compute the corresponding plaintext M.

One may formalize the task faced by this adversary as the RSA Problem: The RSA Problem: Given an RSA public key (n, e) and a ciphertext \(C = M^e\!\!\pmod{n}\),...

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Authors
  1. Ronald L. Rivest
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  2. Burt Kaliski
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Editor information

Editors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

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© 2005 International Federation for Information Processing

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Rivest, R.L., Kaliski, B. (2005). RSA Problem. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_363

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