# Are there...one-way encryption algorithms

Bodo Moeller moeller at cdc.informatik.tu-darmstadt.de
Thu Nov 20 09:42:11 EST 2003

```On Tue, Nov 18, 2003 at 09:19:48AM -0800, Anton Stiglic wrote:
> "David Wagner" <daw at taverner.cs.berkeley.edu>:
>> martin f krafft  wrote:

>>> it came up lately in a discussion, and I couldn't put a name to it:
>>> a means to use symmetric crypto without exchanging keys:
>>>
>>>  - Alice encrypts M with key A and sends it to Bob
>>>  - Bob encrypts A(M) with key B and sends it to Alice
>>>  - Alice decrypts B(A(M)) with key A, leaving B(M), sends it to Bob
>>>  - Bob decrypts B(M) with key B leaving him with M.
>>>
>>> Are there algorithms for this already? What's the scheme called?

>> It's called Pollig-Hellman.

> If I'm not mistaken you are wrong.  Pohlig-Hellman proposed an encryption
> scheme based on discret log, the description of the OP was for a
> key transport protocol.
> In Pohlig-Hellman, what you do is have Alice and Bob share secret
> keys k and d such that k*d == 1 mod (p-1), where p is some prime.
> To encrypt a message M Alice computes M^k mod p, and Bob
> can decrypt by computing (M^k)^d mod p == M mod p.
[...]
> The algorithm that was described by the OP is really Shamir's
> three-pass algorithm, also known as Shamir's no-key protocol.

The Pohlig-Hellman cipher is the modular scheme that you describe, but
observe there is a connection to the protocol above: that protocol
works only if encryption and decryption has a certain commutativity
property (decrypting  B(A(M))  with key  A   must leave  B(M),  not
just some  A^-1(B(A(M)))  that might look entirely different), and
the Pohlig-Hellman cipher has this property.

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