#  in cryptography@c2.net mail archive

## Re: The Pure Crypto Project's Hash Function

#### daemon@ATHENA.MIT.EDU (Ronald L. Rivest)Sun May 4 11:30:03 2003

```X-Original-To: cryptography@metzdowd.com
X-Original-To: cryptography@metzdowd.com
Date: Sun, 04 May 2003 10:44:03 -0400
To: Ralf Senderek <ralf@senderek.de>,
tom st denis <tomstdenis@yahoo.com>
From: "Ronald L. Rivest" <rivest@mit.edu>
Cc: <cryptography@metzdowd.com>,

At 02:57 AM 5/4/2003, Ralf Senderek wrote:
>...
>
>Does the list know of any hash based on Modexp with a better reputation
>than mine, I'd be happy to know.
>
>Ralf.

Adi Shamir once proposed the following hash function:

Let n = p*q be the product of two large primes, such that
factoring n is believed to be infeasible.

Let g be an element of maximum order in Z_n^* (i.e. an
element of order lambda(n) = lcm(p-1,q-1)).

Assume that n and g are fixed and public; p and q are secret.

Let x be an input to be hashed, interpreted as a
non-negative integer.  (Of arbitrary length; this may be
considerably larger than n.)

Define hash(x) = g^x (mod n).

Then this hash function is provably collision-resistant, since
the ability to find a collision means that you have an x and
an x' such that

hash(x) = hash(x')

which implies that

x - x' = k * lambda(n)

for some k.  That is a collision implies that you can find a
multiple of lambda(n).  Being able to find a multiple of lambda(n)
means that you can factor n.

I would suggest this meets the specs of your query above.

Cheers,
Ron Rivest

Ronald L. Rivest
Room 324, 200 Technology Square, Cambridge MA 02139
Tel 617-253-5880, Fax 617-258-9738, Email <rivest@mit.edu>

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