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# Cayley-Purser Algorithm

The Cayley-Purser algorithm is a public-key cryptography algorithm that relies on the fact that matrix multiplication is not commutative. It was devised by Sarah Flannery (then age 16), inspired by ideas of Michael Purser, for a Young Scientist competition in 1998. Flannery named the algorithm after Purser and Arthur Cayley (the inventor of matrices).

The Cayley-Purser algorithm uses only modular matrix multiplication instead of modular exponentiation, and is much faster than other public-key algorithms for large moduli (e.g., about 20 times faster than RSA encryption for a 200-digit modulus). While the algorithm at first seemed to be as secure as RSA, it was subsequently shown that messages encrypted with this algorithm can be readily decrypted by knowledge of public data alone.

Setup for the Cayley-Purser algorithm proceeds as follows.

1. Choose two large primes and , and let . The primes and should both be of the form , where is also prime.

2. Choose matrices and at random from (the general linear group of invertible matrices whose entries are integers modulo ) so that .

3. Choose and let .

4. Let .

5. Publish , , , and . Keep secret the matrix , which is needed for decryption.

To encrypt, use the following procedure.

1. Represent the message as a matrix with entries in .

2. Choose at random.

3. Let .

4. Let .

5. Encrypt as .

6. Transmit the pair .

Decryption is performed as follows.

1. Let .

2. Then . Since and commute, we have

 (1) (2) (3)

Because is the product of two large, unknown primes, it is computationally difficult to determine the original matrix via . In addition, solving

 (4)

for is intractable since, as a result of the way and are chosen, the order of the center of in will be at least with probability .

However, it is not necessary to know to decrypt messages; knowledge of a multiple of is sufficient, since if , it follows that

 (5)

If the matrix is derogatory, (that is, if is reduced modulo or modulo , the result is a scalar multiple of the identity), then can be factored as . With a factorization of in hand, determining the solution to in is not difficult.

If is nonderogatory, then a multiple of of can be determined as follows. Since , then by the Cayley-Hamilton theorem, there are integers and so that . Consequently, there is a matrix for some as yet unknown value of .

Also observe that

 (6)

so

 (7)

Thus,

 (8) (9) (10)

This allows , and hence , to be determined. The matrix can be used in place of in the decryption step.

Public-Key Cryptography, RSA Encryption

This entry contributed by Scott Sutherland

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## References

Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, 2000.

## Referenced on Wolfram|Alpha

Cayley-Purser Algorithm

## Cite this as:

Sutherland, Scott. "Cayley-Purser Algorithm." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Cayley-PurserAlgorithm.html