“RSA Encryption”的概念、定义、习题解题方法及翻译-数学辞典

单词

RSA Encryption

释义

RSA Encryption

A Public-Key Cryptography Algorithm which uses Prime Factorization as the Trapdoor Function. Define

(1)

for

and

Primes. Also define a private key

and a public key

such that

(2)

(3)

where

is the Totient Function.

Let the message be converted to a number . The sender then makes and public and sends

(4)

To decode, the receiver (who knows

) computes

(5)

since

is an Integer. In order to crack the code,

must be found. But this requires factorization of

since

(6)

Both

and

should be picked so that

and

are divisible by large Primes, since otherwise thePollard p-1 Factorization Method or Williams p+1 Factorization Method potentially factor

easily. It is also desirable to have

large and divisible by large Primes.

It is possible to break the cryptosystem by repeated encryption if a unit of has smallOrder (Simmons and Norris 1977, Meijer 1996), where is the Ring ofIntegers between 0 and under addition and multiplication (mod ). Meijer (1996) shows that``almost'' every encryption exponent is safe from breaking using repeated encryption for factors of the form

			(7)
			(8)

where

			(9)
			(10)

and

, and

are all Primes. In this case,

			(11)
			(12)

Meijer (1996) also suggests that

and

should be of order 10⁷⁵.

Using the RSA system, the identity of the sender can be identified as genuine without revealing his private code.

See also Public-Key Cryptography

References

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 166-173, 1985.

Meijer, A. R. ``Groups, Factoring, and Cryptography.'' Math. Mag. 69, 103-109, 1996.

Rivest, R. L. ``Remarks on a Proposed Cryptanalytic Attack on the MIT Public-Key Cryptosystem.'' Cryptologia 2, 62-65, 1978.

Rivest, R.; Shamir, A.; and Adleman, L. ``A Method for Obtaining Digital Signatures and Public Key Cryptosystems.'' Comm. ACM 21, 120-126, 1978.

RSA Data Security. A Security Dynamics Company. http://www.rsa.com.

Simmons, G. J. and Norris, M. J. ``Preliminary Comments on the MIT Public-Key Cryptosystem.'' Cryptologia 1, 406-414, 1977.

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