Claim indefinite for variables with no meaning attached (Ex parte Nishioka)

Ex parte Nishioka
(Appeal 2009-001507; Appl. No. 10/046,224; Tech. Center 2400)
Decided January 12, 2010

This patent application is related to encryption, so I would expect math to be involved. But look at the claim — it appears to me to be nothing but math: 

I'm not afraid of math, but this is too much for me to think about (at least in my free time). So I'm not going to discuss the substance of these claims. But I thought the indefiniteness rejection might be a good lesson for those who deal with math-intensive applications from time to time.

The Examiner viewed the claims as a laundry list of elements, with no relationship. Worse, the elements themselves were "undefined."

[The independent claims] recite the limitations "a key generation step of generating a secret-key" and "a public-key" with a list of undefined elements in between without a relation of how are they being used to generate a key or how are they obtained/selected. The relationship between the elements listed and how they are used to produce a key is not clear. Examiner contrasts the language found in the instant application with the language found in the prior art's claims, where the variables are clearly defined. A clear definition of the variables would overcome this 112 rejection.
(Examiner's Answer, p. 25.)

In response, the Applicant simply asserted that the elements in the claim are well known to one of ordinary skill in the art, then "invite[d] the Board to provide a statement of how to overcome this rejection."

The Board affirmed the indefiniteness rejection, noting that variables are not definite unless they have meaning:

This is an art of very high complexity (FF #3), and the practitioner in such an art is assumed to have advanced math skills. But the attachment of meaning to the variables expressed in the equations of the claims, either through definitions in the specification or in the claims themselves, adds necessary clarity to the claims to be able to properly appreciate the metes and bounds of the protected invention. Appellants have not done this. The Examiner’s contrast of the instant claims to those in the prior art was insightful.
(Decision, p. 9.)

Although the Board didn't come right out and say it, I'd say the reference to the claims in the prior art was a strong hint to the Applicant as to the type of claims that would comply with §112 ¶2nd. I've reproduced the claims from the prior art reference below.

You can see that while variables are used, so are descriptions of the variables. Also, even mathematical functions are described in words (i.e., product of first base-group-number raised to the power of a first monotone function of said first exponent-number).

I think the lesson here is that when dealing with math-intensive subject matter, you should think carefully about simply pasting equations into the claims.