Debunking the PGP backdoor myth for good. [was RE: Hypothesis: PGP backdoor (was: A security bug in PGP products?)]

[Leichter, Jerry]

| On 8/28/06, Ondrej Mikle <ondrej.mikle at gmail.com> wrote:
| > Take as an example group of Z_p* with p prime (in another words: DLP).
| > The triplet (Z, p, generator g) is a compression of a string of p-1
| > numbers, each number about log2(p) bits.
| 
| Pardon my mathematical ignorance, but isn't Z just a notation to indicate
| a ring, as opposed to a parameter that you'd have to store?
Z is universally used to represent the integers.  (From Zahlen, German
for numbers.)  In printed mathematics, Z used this way is taken from a
special "blackboard bold" font.  A common representation uses two
parallel strokes for the Z, with somewhat thickened horizontal bars.
(Back when math was typed on a typewriter, you produced this by typing
Z, backspacing almost but not quite all the way, then typing it again.)
The same font is also used for the reals (R), rationals (Q - from
quotient?) and the complexes (C).  The Hamiltonians are less common, but
you'll sometimes see an H from this font to name them.  N is sometimes
used for the natural numbers (positive integers).  (The naturals are not
much used beyond elementary-school texts....)  The other letters in the
font have no universal meaning, but get used in specialized areas.  I
think I've seen a black-board bold A used for an affine space, for
example.

In all cases, the "usual" operations are assumed, so R is the reals as a
complete ordered field, Z is the ring of integers under the usual
addition and multiplication (with the usual ordering, though there is no
common formal name I know of for a ring with an associated ordering),
and so.

There are a bunch of associated notions, like Z_n (_ for subscript - TeX
notation) for the group of integers mod n under addition.  When n\p is a
prime, Z_p^* (^ for superscript) for the group of integers 1..p mod p
under multiplication.  Z_n is actually a ring under addition and
multiplication mod n, and Z_p a field, and where appropriate, they are
taken to be so.  Q_p is the p-adics, but that's getting specialized.

In ASCII, we don't of course have blackboard bold fonts, but Z is mainly
taken to be the integers, and Z_p is universally taken to be the
integers mod p, in discussions even vaguely related to integer
properties.  R and the others are less commonly used, and you'd have to
understand the context.

Mr. Mikle's notation is ... a bit odd.  What else might one conceivably
substitute for the integers in (Z, p, generator g)?  If it has to be the
integers, why describe this as a triplet?
							-- Jerry

---------------------------------------------------------------------
The Cryptography Mailing List
Unsubscribe by sending "unsubscribe cryptography" to majordomo at metzdowd.com