December 2003 - cypherpunks-legacy

Freedom, Starvation, and Uncoerced Relationships (fwd)
by Jim Choate 17 Dec '03

17 Dec '03

Forwarded message: > Date: Sat, 10 Jan 1998 11:15:42 -0800 > From: Tim May <tcmay(a)got.net> > Subject: Freedom, Starvation, and Uncoerced Relationships > >Since when has a crypto anarchist been a market anarchist? > > And how else could it be? Easily. The acceptance of freedom of speech is not equivalent to the acceptance to spend and earn money freely and without regulation. Speech and money are *not* equivalent. This is as specious as Vulis' argument that these positions are equivalent to pregnancy in the logical realm, in short you are or aren't. The reality is that few people, other than statists or extremists look at the world let alone their personal beliefs in that simplistic fashion. I believe in the unregulated exercise of speech, including the dissemination and use of crypto technology. To do otherwise implies some sort of ownership of the individual by the society doing the regulating. Clearly an incorrect conclusion. The ability of those same groups to spend their money as they see fit is not supportable by that same logic. Groups must have regulated monetary systems or else they collapse because of the monopolization and therefore loss of vitality of markets. To believe otherwise is to accept the premise that these monopolies could somehow buy an individuals rights. Clearly a result even a market anarchist can't accept. ____________________________________________________________________ | | | Those who make peaceful revolution impossible will make | | violent revolution inevitable. | | | | John F. Kennedy | | | | | | _____ The Armadillo Group | | ,::////;::-. Austin, Tx. USA | | /:'///// ``::>/|/ http://www.ssz.com/ | | .', |||| `/( e\ | | -====~~mm-'`-```-mm --'- Jim Choate | | ravage(a)ssz.com | | 512-451-7087 | |____________________________________________________________________|

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The name "Crypto Kong" (fwd)
by Jim Choate 17 Dec '03

17 Dec '03

Forwarded message: > Date: Sat, 10 Jan 1998 11:18:17 -0800 (PST) > From: "James A. Donald" <jamesd(a)echeque.com> > Subject: The name "Crypto Kong" > I have received some negative feedback about the name > "Crypto Kong". > > Two people have complained that it is unprofessional > sounding. Two people? That's it? > This is not necessarily grounds for alarm. One reason the > name irritates people is that it sticks in the mind like a > bad song, which was of course an important reason for > choosing it. No one is likely to say "Hey, I saw this plug > for some digital signing tool, but I can't remember the > name." When I think of it I think of 'Donkey Kong'. > What do you think? Unless the loss would result in ten's of thousands of dollars of income leave it. If it would effect a large sales market it under a more acceptable name to the business community. Nothing unethical about selling the same product under two different names... ____________________________________________________________________ | | | Those who make peaceful revolution impossible will make | | violent revolution inevitable. | | | | John F. Kennedy | | | | | | _____ The Armadillo Group | | ,::////;::-. Austin, Tx. USA | | /:'///// ``::>/|/ http://www.ssz.com/ | | .', |||| `/( e\ | | -====~~mm-'`-```-mm --'- Jim Choate | | ravage(a)ssz.com | | 512-451-7087 | |____________________________________________________________________|

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Nervous Nellies at the PO
by Gary Harland 17 Dec '03

17 Dec '03

Came across this by chance...having never heard of this 'disgruntled' web site, I'm curious as to how the PO managed to learn about the offending story...if they fired him because of an explicity fictional article expressing a fantasy shared by vast hordes of Americans then it kind of looks like the Post Office 'went postal' on one of their employees. > DISGRUNTLED POSTAL WORKER > > A South Bay postal worker has been fired for a fictional article he > got published on the Internet, in which he depicted a worker who was > so fed up with conditions in his San Jose office that he pulled out a > gun and shot his dictatorial and much despised supervisor. The article > appeared in December in the on-line publication Disgruntled, which > bills itself as the business magazine for people who work for a > living. > > The story, "Scrooged Again," appeared to have struck a nerve with his > employer, who informed him that he would be removed from his job at > the postal service effective Jan. 27 because of "unacceptable and > disrespectful conduct." > > The Web page site for Disgruntled is: http://www.disgruntled.com > ----------------------------------------------------------------- foggy(a)netisle.net lat:47d36'32" long:122d20'12" "Rather perish than hate and fear, and twice rather perish than make oneself hated and feared." -F. Nietzche- -----------------------------------------------------------------

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Re: Spam
by lcs Mixmaster Remailer 17 Dec '03

17 Dec '03

>I've got a question for y'all. > >Some idiot finally sent me a junk e-mail message that >I couldn't do anything about with what I have the >knowledge to do -- reading the headers turned up only >one ISP, which was apparently owned by the spammer. > >I'm hoping you guys would know where I could find out >more about this individual -- I'm hoping that he does >in fact buy his service from someone else, and if so, >I'm not sure how to find that out. If he doesn't, is >there anything I can do? His traffic has to be routing through somebody or multiple somebodies. Traceroute it and complain to the upstream site. Unless they're cretins like CIX they'll kill his connection.

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LUC Public Key Crypto...
by Jim Choate 17 Dec '03

17 Dec '03

Hi, In the process of doing some research on Gauss I stumbled across this... ____________________________________________________________________ | | | Those who make peaceful revolution impossible will make | | violent revolution inevitable. | | | | John F. Kennedy | | | | | | _____ The Armadillo Group | | ,::////;::-. Austin, Tx. USA | | /:'///// ``::>/|/ http://www.ssz.com/ | | .', |||| `/( e\ | | -====~~mm-'`-```-mm --'- Jim Choate | | ravage(a)ssz.com | | 512-451-7087 | |____________________________________________________________________| Forwarded message: > Dr. Dobb's Web Site > > LUC PUBLIC-KEY ENCRYPTION > > > > A secure alternative to RSA > > Peter Smith > > Peter has worked in the computer industry for 15 years as a > programmer, analyst, and consultant and has served as deputy editor of > Asian Computer Monthly. Peter's interest in number theory led to the > invention of LUC in 1991. He can be reached at 25 Lawrence Street, > Herne Bay, Auckland, New Zealand. > > > _________________________________________________________________ > > According to former NSA director Bobby Innman, public-key cryptography > was discovered by the National Security Agency in the early seventies. > At the time, pundits remarked that public-key cryptography (PKC) was > like binary nerve gas--it was potent when two different substances > were brought together, but quite innocuous in its separate parts. > Because the NSA promptly classified it, not much was known about PKC > until the mid-seventies when Martin Hellman and Whitfield Diffie > independently came up with the notion and published papers about it. > > Traditional cryptographic systems like the venerable Data Encryption > Standard (DES) use the same key at both ends of a message > transmission. The problem of ensuring correct keys leads to such > expensive expedients as distributing the keys physically with trusted > couriers. Diffie and Hellman (and the NSA) had the idea of making the > keys different at each end. In addition to encryption, they envisioned > this scheme would also lead to a powerful means of source > authentication known as digital signatures. > > RSA, developed in 1977, was the first reliable method of source > authentication. The RSA approach (patented in the early eighties) > initiated intense research in "number theory," one of the most > recondite areas of mathematics. Although C.F. Gauss studied this topic > in the early 1800s (referring to it then as "higher arithmetic"), very > little real progress has been made in solving the problem of factoring > since then. The means available today are essentially no better than > exhaustive searching for prime factors. In terms of intractability > theory, however, no one has yet proved that the problem is > intractable, although researchers believe it to be so. > > > > The RSA Algorithm > > > > RSA works by raising a message block to a very large power, then > reducing this modulo N, where N (the product of two large prime > numbers) is part of the key. Typical systems use an N of 512 bits, and > the exponent to which blocks are raised in decryption is of the same > order. An immediate problem in implementing such a system is the > representation and efficient manipulation of such large integers. > (Standard microprocessors don't really have the power to handle normal > integer sizes and functions; even numeric coprocessors are inadequate > when integers of this size are involved.) > > RSA has dominated public-key encryption for the last 15 years as > research has failed to turn up a reliable alternative--until the > advent of LUC. Based on the same difficult mathematical problem as > RSA, LUC uses the calculation of Lucas functions instead of > exponentiation. (See text box entitled, "How the Lucas Alternative > Works.") > > Because we're working in the area of mathematics, we can formally > prove that LUC is a true alternative to RSA. Furthermore, we can show > that a cipher based on LUC will be at least as efficient. More > importantly, we can show that LUC is a stronger cipher than RSA. The > reason is that under RSA, the digital signature of a product is the > product of the signatures making up the product; in mathematical > terms, M{e}L{e}=(ML){e}. This opens RSA to a cryptographic attack > known as adaptive chosen-message forgery. Ironically, this is outlined > in a paper co-authored by Ron Rivest (the "R" in RSA). LUC is not > multiplicative and therefore not susceptible to this attack. Using > Lucas functions, V[e](M,1)V[e](L,1) is not equal to V[e](ML,1). In > other words, the use of exponentiation leads to RSA being > multiplicative in this way, while LUC's use of Lucas functions avoids > this weakness. > > Choosing the Algorithms > > > > Lucas functions have been studied mainly in relation to primality > testing, and it was to these sources we turned when researching > efficient algorithms for implementing LUC. For given parameters, the > Lucas functions give rise to two series, U[n] and V[n]. The first > algorithm (see Listing One, page 90) calculated both, even though we > were only interested in V[n]. It was only in a paper on factoring > integers that we found a means of calculating V[n] alone (see Listing > Two, page 90). The pseudocode examples show that both algorithms have > two phases: The work done when the current bit is a 0 is half the work > necessary when the current bit is a 1. > > More Details. > > Typically, in systems like LUC the exponent used for encryption is a > much smaller integer than that used for decryption. A commonly chosen > encryption exponent is the prime number 65,537. This is a good choice > for fast encryption as all but 2 of the 17 bits are 0s. We have no > such control over the decryption exponent, but there is a way of > halving the work, and thus, of introducing a limited degree of > parallelism into the calculation. > > Since LUC is a public-key cryptosystem, we can always assume that the > possessor of the private decrypting keys knows the two primes (p and > q) which make up the modulus, N. Consequently, we can reduce the > exponent and message with respect to the two primes, in each case at > least halving the amount of work. At the end of the calculation with > respect to the primes, we bring the results together to produce the > final plain text (see Listing Three, page 90). > > Large-integer Arithmetic > > > > There's really only one source of information about large-integer > arithmetic: Knuth's The Art of Computer Programming. We found that > almost every time we referred to his book, we came up with some new > angle or way of tweaking some extra performance out of our code. > > We decided to represent the large integers as 256-byte arrays, with > the low byte giving the length (in bytes) of the integer. For > instance, the 8-byte hexadecimal number 1234567890ABCDEF would appear > in a file view as 08 EF CD AB 90 78 56 34 12. These arrays became a > Pascal-type har (for hexadecimal array). We can store integers of over > 600 decimal digits in our hars, but because the hars must be able to > hold the results of a multiplication, we are limited to manipulating > integers up to 300 decimal digits in length. > > Implementation of addition, subtraction, and multiplication went quite > smoothly; implementation of division took more effort. (We took > comfort in not being the first to encounter problems with division. > Lady Ada Lovelace, the first computer programmer, said, "I am still > working at some most entangled notations of division, but see my way > through them at the expense of heavy labor, from which I shall not > shrink as long as my head can bear it.") We tried various methods, > including one based on Newton which calculated the inverse of the > divisor and then multiplied. (See Knuth's discussion.) We finally > opted for Knuth's Algorithm D, despite his warning that it contained > possible discontinuities. At that stage, we were working on a 16-bit > 80286 PC; see Listing Four, page 90. > > Of course there was much more than the division routine to consider, > but we found that it was the critical routine in terms of getting LUC > to run at a reasonable speed. Once we had upgraded to an 80386, we > converted to a full 32-bit implementation. The assembler code for the > division (still Algorithm D) is given in Listing Five (page 91). > Although space constraints prevent a complete presentation of the > code, suffice to say that we have been able to achieve a > signing/decryption speed on a modulus of 512 bits of over 200 bits per > second (33-MHz 80386, 0 wait states). > > Other Issues > > > > Central to any cryptographic system are keys. In LUC, if an adversary > is able to find p and q, the prime factors of modulus N, then all > messages sent with N can be either read in the case of encryption or > forged in the case of signing. > > Since the days of Gauss, research on factoring has come up with > various so-called "aleatoric" methods of factoring some numbers. These > methods are like cures for poison ivy: numerous, and occasionally > efficacious. One old method, found by Pierre Fermat, is very quick at > factoring some types of composite numbers. If N is the product of two > primes which are close together, then it can be easily factored. For > example, if p=1949, and q=1951, then N=3802499. Taking the square root > of N, we find that it is approximately 1949.999. Adding 1 to the > integral part of this (giving 1950), we square this, giving 3802500. > If we now subtract N from this square, we get a difference of 1, which > is the square of itself. This means that N has been expressed as the > difference of two squares. As we learned in high school, x{2}-y{2} = > (x-y)(x+y), and so we obtain the two factors. > > Fermat's method works whenever the ratio of the factors is close to an > integer. (Note that the ratio is close to 1 in the above discussion.) > This attack, as cryptographers call methods used to break a cipher, > has to be guarded against in generating the modulus N. > > Another guard is that neither (p + 1) and (q + 1) nor (p - 1) and (q - > 1) should be made up of small prime factors. There are many other > guards of varying degrees of importance, but the entire area needs > consideration depending on the level of security required, and how > long the keys are meant to last. > > The basic idea behind LUC is that of providing an alternative to RSA > by substituting the calculation of Lucas functions for that of > exponentiation. While Lucas functions are somewhat more complex > mathematically than exponentiation, they produce superior ciphers. > > This substitution process can be done with systems other than the RSA. > Among these are the Hellman-Diffie-Merkle key exchange system (U.S. > Patent number 4,200,770), the El Gamal public-key cryptosystem, the El > Gamal digital signature, and the recently proposed Digital Signature > Standard (DSS), all of which use exponentiation. > > The nonmultiplicative aspect of Lucas functions carries over, allowing > us to produce alternatives to all these. In the case of the DSS, Lucas > functions allow us to dispense with the one-way hashing cited (but not > specified) in the draft standard. > > A New Zealand consortium has been set up to develop and license > systems based on LUC, which is protected by a provisional patent. For > more information, contact me or Horace R. Moore, 101 E. Bonita, Sierra > Madre, California 91024. > > References > > > > Athanasiou, Tom. "Encryption Technology, Privacy, and National > Security." MIT Technology Review (August/September, 1986). > > Diffie, W. and M.E. Hellman. "New Directions in Cryptography." IEEE > Transactions on Information Theory (November, 1976). > > El Gamal, Taher. "A Public Key Cryptosystem and a Signature Scheme > Based on Discrete Logarithms." IEEE Transactions on Information Theory > (July, 1985). > > Gauss, C.F. "Disquisitiones Arithmeticae," Article 329. > > Goldwasser, S., S. Micali, and R. Rivest. "A Digital Signature Scheme > Secure Against Adaptive Chosen Message Attack." SIAM J. COMPUT (April, > 1988). > > Kaliski, Burton S., Jr. "Multiple-precision Arithmetic in C." Dr. > Dobb's Journal (August, 1992). > > Knuth, D.E. The Art of Computer Programming: Volume II: Semi-Numerical > Algorithms, second edition. Reading, MA: Addison-Wesley, 1981. > > Schneier, Bruce. "Untangling Public Key Cryptography." Dr. Dobb's > Journal (May, 1992). > > Williams, H.C. "A p + 1 method of factoring." Mathematics of > Computation (vol. 39, 1982). > > How the Lucas Alternative Works > > > > As with RSA encryption, use of the Lucas alternative involves two > public keys: N and e. The number N is assumed to be the product of two > large (odd) prime numbers, p and q. Encryption and decryption of a > message is achieved using Lucas sequences, which may be defined as > shown in Example 1. Note that P and Q are integers. > > If a message P is to be sent, it is encoded as the residue P1 modulo N > of the eth term of the Lucas sequence V[n](P,1), and then transmitted. > The receiver uses a secret key d (based on the prime factorization of > N) to decode the received message P1, by taking the residue modulo N > of the dth term of the Lucas sequence V[n](P1,1). The secret key d is > determined so that V[d](V[e](P,1),1) = P modulo N, ensuring the > decryption of the received message P1 as P. The existence of such a > key d is based on the following theorem. > > Theorem > > > > Suppose N is any odd positive integer, and P is any positive integer, > such as P{2}-4 is coprime to N. If r is the Lehmer totient function of > N with respect to D = P{2}-4 (see Example 2), then V[mr+1](P,1)=P > modulo N for every positive integer m. The condition that P{2}-4 be > coprime to N is easily checked, as P{2}-4=(P+2)(P-2). Also, because > V[d](V[e](P,1),1)=V[de](P,1), according to Example 4(e), the key d may > simply be chosen so that de=1 modulo r. > > The Lehmer Totient Function > > > > Suppose P and Q are integers, and a and b are the zeros of X{2}-Px+Q > (so that P = a+b while Q = ab). Also, let D be the discriminant of > x{2}-Px+Q. That is, D = P{2}-4Q = (a-b){2}. > > The Lucas sequences U[n] = U[n] (P,Q) and V[n] = V[n] (P,Q) are > defined for n = 0,1,2, and so on by the equation in Example3. > > In particular, U[0] = 0, U[1] = 1, and then U[n+1] = PU[n] - QU[n-1] > (for n = 1,2,3,...), while V[0] = 2, V[1] = P, and similarly V[n+1]= > PV[n]-QV[n-1] (for n = 1,2,3,...). These sequences satisfy a number of > identities, including the following which may be simply obtained from > the definitions in Example 4. > > Next, suppose N is any positive integer, and let r be the Lehmer > totient function of N with respect to D = P{2}-4Q, defined the same > way as in the statement of the theorem. In the special case where N is > an odd prime p, the Lehmer totient function of p with respect to D is > the number given by the equation in Example 5(a). In this case, the > Lucas-Lehmer theorem states that if p does not divide Q then the > equation in Example 5(b) holds true. > > Example of LUC > > > > Let N = pxq = 1949x2089=4071461, and P = 11111, which equals the > message to encrypt/decrypt. The public keys will be e and N; the > private key will be d. First, calculate r, the Lehmer totient function > of P with respect to N. To do this we need to calculate the Legendre > of p and q. Let D = p{2}-4; then (D/1949) =-1 and (D/2089)=-1 are the > two Legendre values. Hence r is the least common multiple of 1949 + 1 > and 2089 + 1; see Example 6(a). Choosing e = 1103 for our public key, > we use the Extended Euclidean Algorithm to find the secret key d, by > solving the modular equation ed = 1 mod r. d turns out to equal 24017. > > To encrypt the message 11111, we make the calculation shown in Example > 6(b). To decrypt the encrypted message, we calculate as in Example > 6(c). --P.S. > > > _LUC PUBLIC-KEY ENCRYPTION_ > by Peter Smith > > > [LISTING ONE] > > { To calculate Ve(P,1) modulo N } > Procedure LUCcalc; > {Initialise} > BEGIN > D := P*P - 4; ut := 1; vt := P; u := ut; v := vt; > If not odd(e) then BEGIN u := 0; v := 2; END; > e := e div 2; > {Start main} > While e > 0 do > BEGIN > ut := ut*vt mod N; vt := vt*vt mod N; > If vt < 3 then vt := vt + N; > vt := vt - 2; > If odd(e) then > BEGIN > c := (ut*v + u*vt) mod N; > v := (vt*v + D*u*ut) mod N; > If odd(v) then v := v + N; v := v/2; > If odd(c) then c := c + N; u := c/2; > END; > e := e div 2; > END; > END; {LUCcalc} > > { The required result is the value of v.} > > > > > > > [LISTING TWO] > Pseudocode for calculating Lucas Functions > > Procedure wiluc { V = V(M) Mod N, the Mth Lucas number(P,1) } > Var > V,Vb,P,Vf,N,M,NP, Vd, Vf : LargeInteger ; > carry, high_bit_set : boolean ; > bz : word ; > BEGIN > Va := 2 ; { V[0] } Vb = P ; { V[1] } > NP := N - P; bz := bits(M) -1 ; { test bits from high bit downwards } > For j := 1 to bz do > BEGIN > Vc := Vb * Vb; Vf = Vc ; If Vf < 2 then Vf := Vf + N > Vf := Vf - 2; Vd := Va * Vb > { Vc := V, Vd := V*Vb, Vf := V-2} > If high_bit_set Then > BEGIN > Vb := P * Vc; If Vb < Vd then Vb := Vb + N; Vb := Vb - Vd; > If Vb < P then Vb := Vb + N; Vb := Vb - P; Va := Vf > END ; > Else BEGIN { "even" ie high bit not set } > Va := Vd; If Va < P then Va := Va + N; Va := Va - P; > Vb := Vf; > END ; > High_bit_set := next_bit_down(M); > {This boolean function determines the setting of the next bit down} > Va := Va Mod N; Vb := Vb Mod N > END ; { for j to bz } > END ; {wiluc} > > > > > > > [LISTING THREE] > > { Pseudocode for splitting decryption/signing over p and q > (N = p*q) } > Procedure hafluc ( var s,p,q,m,e : LargeInteger ; qix : word ) ; > var ep,emq, > temp,pi,qi, > b,n,pa,qa : LargeInteger ; > > { This procedure applies only to decipherment and signing, where the primes > making up the modulus N ( = p * q) are known (or can be easily deduced, > since both keys are known). Applying it allows us to halve the amount of > work. Encipherment is usually done with a small key - standard is 65537. } > Begin > Qpr (pa,qa,p,q,m,qix ) ; {} {assumes qix already calculated } > ep = e ; ep = ep Mod pa > emq = e ; emq = emq Mod qa > mp = m ; mp = mp Mod p > mq = m ; mq = mq Mod q > wiluc(q2,mq,emq,q) ; wiluc(p2,mp,ep,p) ; > if p2 < q2 then > Begin > temp = q q = p p = temp > temp = q2 q2 = p2 p2 = temp > End ; > temp = p2 temp = temp - q2 > n = p * q > { Solve with Extended Euclidean algorithm qi = 1/q Mod p. The algorithm > for the Extended Euclidean calculation can be found in Knuth. } > r = temp * p > r = r mod N > s = r * qi > s = s Mod n > s = s + p2 > End ; { hafluc } > Procedure SignVerify ; > Begin > h4 = 4 > p = large prime... > q = large prime... > n = p * q > bz := bits(n) ; > {write(cf,' generate 4 keysets (d,e) for p1,q1') ;} > { > qix table for T[qix] > Convention for qix > This calculation is explained below. > Lehmer totient qix Legendre values for p and q > i.e. T[qix] = LCM > (p - 1),(q - 1) 1 1 1 > (p - 1),(q + 1) 2 1 -1 > (p + 1),(q - 1) 3 -1 1 > (p + 1),(q + 1) 4 -1 -1 > e = encryption key, small prime eg 65537 > mu = message as large integer less than n > Solve e * d[qix] = 1 Mod T[qix] using Extended Euclidean Algorithm > where T[qix] is lcm(p1,q1), the Lehmer totient function of N > with repect to mu, according to the above table. > This gives 4 possible values of d, the decryption/signing key. > The particular value used depends on the message mu, as follows: > Let D = mu2 - 4. Calculate the Legendre values of D with respect to > both p and q. This value is -1 if D is a quadratic non-residue of > p (or q), and equal to 1 if D is a quadratic residue of p (or q). > N.B. This part is the most difficult part of LUC! Take care. > > Signing (Deciphering): > hafluc (a,pu,qu,mu,d,qix) > > Verifying (Enciphering): > Use Wiluc. > End. > > > > > > > [LISTING FOUR] > > Algorithm D in 32-bit Intel assmbler > Author: Christopher T. Skinner > Short version of Mod32.Txt with scalings just as comments > Modulus routine for Large Integers > u = u Mod v > Based on: > D.E.Knuth The Art of Computer Programming > Vol 2 Semi-Numerical Algorithms 2ed 1981 > Algorithm D page 257 > We use a Pascal Type called "har" ( for "hexadecimal array") > Type > har = Array[0..255] of byte ; > Var u,v : har ; > Note that u[0] is the length of u and that the > integer begins in u[1] > It is desirable that u[1] is on a double word boundary. > > ; Turbo Pascal Usage: ( Turbo Pascal v6.0) > ; {$L Mod32a} { contains mod32 far } > ; {$F+} { far pointers } > ; procedure Mod32 ( var u,v : har ) ; > ; Turbo Assembler code: (TASM v2.01)--requires 32-bit chip ie 386 or 486 > ; nb FS and GS can be used as temporary storage. Don't try to use them as > ; segment registers because Windows 3.0 restricts their allowed range, even > ; after you have finished out of Windows. You will hang for sure, unless you > ; have used a well-behaved protected-mode program to reset them, or cold boot. > > Data Segment Word Public Use16 > vdz dw ? ; size v words > va dd ? ; hi dword v > vb dd ? ; 2nd " v > vi dw ? ; ^v[1] > savdi dw ? ; used in addback > Data EndS > > Code Segment Word Public Use16 > Assume cs:Code, ds:Data ,es:Nothing > Public mod32 > ; Pascal Parameters: > u Equ DWord Ptr ss:[bp+10] ; Parameter 1 of 2 (far) > v Equ DWord Ptr ss:[bp+ 6] ; parameter 2 of 2 > uof equ word ptr ss:[bp+10] > vinof equ word ptr ss:[bp+ 6] > > mod32 Proc far > push bp > mov bp,sp > push di > push si > push ds ; save the DS > > ; Before using Mod32 check that: > ; v > 0 > ; v < u u <= 125 words > ; v[0] is a multiple of 4 and at least 8 > ; v[top] >= 80h (may need to scale u & v) > ; make u[0] = 0 Mod 4 (add 1..3 if required) > domod: > ; now point to our v > mov ax,seg v > mov ds,ax > assume ds:Data > mov si, offset v > cld > assume es:Nothing > xor ah,ah > mov al,es:[di] ; ax = size of u in bytes "uz" > mov cx,ax ; cx = uz > mov bx,ax ; bx = uz > mov al,[si] > mov dx,ax ; dx = size v bytes > shr ax,2 > mov vdz,ax ; vdz " dwords vz = 0 mod 4 > sub bx,dx ; bx = uz - vz difference in bytes > mov ax,bx ; ax = uz - vz > sub ax,3 ; ax = uz - vz - 3 -> gs > sub cx,3 ; cx = uz - 3 > add cx,di ; cx = ^top dword u > add ax,di > mov gs,ax ; gs = ^(uz-vz-3) u start (by -4 down to 1) > inc di > mov fs,di ; fs = uf = ^u[1] , end point > inc si > mov vi,si ; vi = ^v[1] > add si,dx > mov eax,[si-4] > mov va,eax ; va = high word of v > mov eax,[si-8] > mov vb,eax ; vb = 2nd highest word v > mov di,cx ; set di to ut , as at bottom of loop > d3: > mov edx,es:[di] ; dx is current high dword of u > sub di,4 > mov eax,es:[di] ; ax is current 2nd highest dword of u > mov ecx,va > cmp edx,ecx > jae aa ; if high word u is 0 , never greater than > div ecx ; mov ebx,eax > mov esi,edx ; si = rh > jmp short ad ; Normal route -- -- -- -- --> > aa: mov eax,0FFFFFFFFh > mov edx,es:[di] ; 2nd highest wrd u > jmp short ac > ab: mov eax,ebx ; q2 > dec eax > mov edx,esi ; rh > ac: mov ebx,eax ; q3 > add edx,ecx > jc d4 ; Knuth tests overflow, > mov esi,edx > ; normal route: > ad: > mul vb ; Quotient by 2nd digit of divisor > cmp edx,esi ; high word of product : remainder > jb d4 ; no correction to quot, drop thru to mulsub > ja ab ; nb unsigned use ja/b not jg/l > cmp eax,es:[di-4] ; low word of product : 3rd high of u > ja ab > d4: ; Multiply & subtract * * * * * * * > mov cx,gs > mov di,cx ; low start pos in u for subtraction of q * v > sub cx,4 > mov gs,cx > xor ecx,ecx > Mov cx,vdz ; word count for q * v > mov si,vi ; si points to v[1] > xor ebp,ebp ; carry 14Oct90 bp had problems in mu-lp > even > ; ** ** ** ** ** ** ** ** > ba: lodsd ; eax <- ds[si] > mul ebx ; dx:ax contains product carry set if dx > 0 > add eax,ebp > adc edx,0 > sub es:[di],eax > adc edx,0 > mov ebp,edx > add di,4 > loop ba ; dec cx , jmp if not 0 > ; .. .. .. . .. .. . .. .. . .. . . .. > sub es:[di],edx > jnc d7 > > mov si,vi ; add back (rare) > mov savdi,di > mov di,gs > add di,4 > clc > mov cx,vdz > bb: lodsd ; eax = ds[si] si + 2 > adc es:[di],eax > inc di > inc di > inc di > inc di > loop bb > xor eax,eax > mov es:[di],eax > mov di,savdi > ; test with: > ; 1,00000000,00000000,00000001/ 80000000,00000000,00000001 > d7: > mov bx,fs ; fs ^u[1] > mov ax,gs ; gs = current u start position > cmp ax,bx ; current - bottom > jb d8 > sub di,4 > jmp d3 > d8: > ; here we would scale u down if it had been scaled up > quex: ; quick exit if v < u > cld ; just in case > pop ds > pop si > pop di > pop bp > ret 8 ; 2 pointers = 4 words = 8 bytes > mod32 EndP ; > Code Ends > End > > > > > > [LISTING FIVE] > > Algorithm D in 16-bit Intel assembler > Author: Christopher T. Skinner > mod16.txt 21 Au8 92 16 bit modulus > ; divm Modulus > Data Segment Word Public > vwz dw ? ; size v words > va dw ? ; hi word v > vb dw ? ; 2nd " v > vi dw ? ; ^v[1] > uf dw ? ; ^u[3] > uz dw ? ; size u byte > vz dw ? ; " v " > ua dw ? ; ^( u[0] + uz - vz -1 ) , mul sub start > ut dw ? ; ^ u[topword] > qh dw ? > uzofs dw ? ; ttt > vzofs dw ? ; ttt > Data EndS > Code Segment Word Public > Assume cs:Code, ds:Data > Public diva > > u Equ DWord Ptr [bp+10] ; ES:DI > v Equ DWord Ptr [bp+6] ; DS:SI > ; NB v Must be Global, DS based... > diva Proc far > push bp > mov bp,sp > push ds > cld ; increment lodsw in mulsub > lds si,v > les di,u > xor ah,ah > mov al,es:[di] ; ax = uz size of u in bytes N.B. uz is not actually used > mov cx,ax ; cx = uz > mov bx,ax ; bx = uz > mov al,ds:[si] > mov dx,ax ; dx = size v bytes > shr ax,1 > mov vwz,ax ; vwz " words > sub bx,dx ; bx = uz - vz difference in bytes > mov ax,bx ; ax = uz - vz > dec ax ; ax = uz - vz - 1 -> ua > dec cx ; cx = uz - 1 > add cx,di ; cx = ^top word u > mov ut,cx ; ut = ^top word u > add ax,di > mov ua,ax ; ua = ^(uz-vz-1) u start (by -2 down to 1) > inc di > mov uf,di ; uf = ^u[1] , end point > inc si > mov vi,si ; vi = ^v[1] > add si,dx > mov ax,ds:[si-2] > mov va,ax ; va = high word of v > mov ax,ds:[si-4] > mov vb,ax ; vb = 2nd highest word v > mov di,cx ; set di to ut , as at bottom of loop > d3: > mov dx,es:[di] ; dx is current high word of u > dec di > dec di > mov ut,di > mov ax,es:[di] ; ax is current 2nd highest word of u > mov cx,va > cmp dx,cx > jae aa ;if high word u is 0 , never greater than > div cx ; > mov qh,ax > mov si,dx ; si = rh > jmp ad ; Normal route -- -- -- -- --> > aa: mov ax,0FFFFh > mov dx,es:[di] ; 2nd highest wrd u > jmp ac > ab: mov ax,qh > dec ax > mov dx,si ; rh > ac: mov qh,ax > add dx,cx > jc d4 ; Knuth tests overflow, > mov si,dx > ad: mul vb ; Quotient by 2nd digit of divisor > cmp dx,si ; high word of product : remainder > jb d4 ; no correction to quot, drop thru to mulsub > ja ab ; nb unsigned use ja/b not jg/l > cmp ax,es:[di-2] ; low word of product : 3rd high of u > ja ab > d4: ; Multiply & subtract * * * * * * * > mov bx,ua > mov di,bx ; low start pos in u for subtraction of q * v > dec bx > dec bx ; > mov ua,bx > Mov cx,vwz ; word count for q * v > mov si,vi ; si points to v[1] > mov bx,qh > xor bp,bp > ; ** ** ** ** ** ** ** ** > ba: lodsw ; ax <- ds[si] si + 2 preserve carry over mul ? > mul bx ; dx:ax contains product carry set if dx > 0 > add dx,bp > xor bp,bp > sub es:[di],ax > inc di > inc di > sbb es:[di],dx > rcl bp,1 > loop ba ; dec cx , jmp if not 0 > ; .. .. .. . .. .. . .. .. . .. . . .. > rcr bp,1 > jnc d7 > > mov si,vi ; add back (rare) > mov di,ua > inc di > inc di > clc > mov cx,vwz > bb: lodsw ; ax = ds[si] si + 2 > adc es:[di],ax > inc di > inc di > loop bb > mov cx,ut > add cx,4 > sub cx,di > shr cx,1 ; word length of u > bc: mov Word Ptr es:[di],0 > inc di > inc di > loop bc ; > dec di ; > dec di ; > clc > d7: > mov ax,uf > cmp ua,ax > jb d8 > dec di ; New these are suspicious, with an add back and a > dec di ; New > jmp d3 > d8: > cld ; just in case > pop ds > pop bp > ret 8 ; 2 pointers = 4 words = 8 bytes ??? > diva EndP ; > Code Ends > End > > > _________________________________________________________________ > > Copyright © 1998, Dr. Dobb's Journal > Dr. Dobb's Web Site Home Page -- Top of This Page > >

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RB_Fermat.html
by Jim Choate 17 Dec '03

17 Dec '03

PIERRE DE FERMAT (1601 - 1665) From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball. While Descartes was laying the foundations of analytical geometry, the same subject was occupying the attention of another and not less distinguished Frenchman. This was Fermat. Pierre de Fermat, who was born near Montauban in 1601, and died at Castres on January 12, 1665, was the son of a leather-merchant; he was educated at home; in 1631 he obtained the post of councillor for the local parliament at Toulouse, and he discharged the duties of the office with scrupulous accuracy and fidelity. There, devoting most of his leisure to mathematics, he spent the remainder of his life - a life which, but for a somewhat acrimonious dispute with Descartes on the validity of certain analysis used by the latter, was unruffled by any event which calls for special notice. The dispute was chiefly due to the obscurity of Descartes, but the tact and courtesy of Fermat brought it to a friendly conclusion. Fermat was a good scholar, and amused himself by conjecturally restoring the work of Apollonius on plane loci. Except a few isolated papers, Fermat published nothing in his lifetime, and gave no systematic exposition of his methods. Some of the most striking of his results were found after his death on loose sheets of paper or written in the margins of works which he had read and annotated, and are unaccompanied by any proof. It is thus somewhat difficult to estimate the dates and originality of his work. He was constitutionally modest and retiring, and does not seem to have intended his papers to be published. It is probable that he revised his notes as occasion required, and that his published works represent the final form of his researches, and therefore cannot be dated much earlier than 1660. I shall consider separately (i) his investigations in the theory of numbers; (ii) his use in geometry of analysis and of infinitesimals; and (iii) his method for treating questions of probability. (i) The theory of numbers appears to have been the favourite study of Fermat. He prepared an edition of Diophantus, and the notes and comments thereon contain numerous theorems of considerable elegance. Most of the proofs of Fermat are lost, and it is possible that some of them were not rigorous - an induction by analogy and the intuition of genius sufficing to lead him to correct results. The following examples will illustrate these investigations. (a) If p be a prime and a be prime to p then a^(p-1) - 1 is divisible by p, that is, a^(p-1) - 1 \equiv 0 (mod p). A proof of this, first given by Euler, is well known. A more general theorem is that a^(\phi(n)) - 1 \equiv 0 (mod n) (mod n), where a is prime to n and \phi(n) is the number of integers less than n and prime to it. (b) An odd prime can be expressed as the difference of two square integers in one and only one way. Fermat's proof is as follows. Let n be the prime, and suppose it equal to x² - y², that is, to (x + y)(x - y). Now, by hypothesis, the only integral factors of n are n and unity, hence x + y = n and x - y = 1. Solving these equations we get x = œ (n + 1) and y = œ (n - 1). (c) He gave a proof of the statement made by Diophantus that the sum of the squares of two integers cannot be of the form 4n - 1; and he added a corollary which I take to mean that it is impossible that the product of a square and a prime of the form 4n - 1 [even if multiplied by a number prime to the latter], can be either a square or the sum of two squares. For example, 44 is a multiple of 11 (which is of the form 4 × 3 - 1) by 4, hence it cannot be expressed as the sum of two squares. He also stated that a number of the form a² + b², where a is prime to b, cannot be divided by a prime of the form 4n - 1. (d) Every prime of the form 4n + 1 is expressible, and that in one way only, as the sum of two squares. This problem was first solved by Euler, who shewed that a number of the form 2^m (4n + 1) can be always expressed as the sum of two squares. (e) If a, b, c, be integers, such that a² + b² = c², then ab cannot be a square. Lagrange gave a solution of this. (f) The determination of a number x such that x²n + 1 may be a square, where n is a given integer which is not a square. Lagrange gave a solution of this. (g) There is only one integral solution of the equation x² + 2 = y³; and there are only two integral solutions of the equation x² + 4 = y³. The required solutions are evidently for the first equation x = 5, and for the second equation x = 2 and x = 11. This question was issued as a challenge to the English mathematicians Wallis and Digby. (h) No integral values of x, y, z can be found to satisfy the equation x^n + y^n = z^n ; if n be an integer greater than 2. This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true. Probably Fermat discovered its truth first for the case n = 3, and then for the case n = 4. His proof for the former of these cases is lost, but that for the latter is extant, and a similar proof for the case of n = 3 was given by Euler. These proofs depend on shewing that, if three integral values of x, y, z can be found which satisfy the equation, then it will be possible to find three other and smaller integers which also satisfy it: in this way, finally, we shew that the equation must be satisfied by three values which obviously do not satisfy it. Thus no integral solution is possible. It would seem that this method is inapplicable to any cases except those of n = 3 and n = 4. Fermat's discovery of the general theorem was made later. A proof can be given on the assumption that a number can be resolved into the product of powers of primes in one and only one way. The assumption has been made by some writers; it is true of real numbers, but it is not necessarily true of every complex number. It is possible that Fermat made some erroneous supposition, but, on the whole, it seems more likely that he discovered a rigorous demonstration. In 1823 Legendre obtained a proof for the case of n = 5; in 1832 Lejeune Dirichlet gave one for n = 14, and in 1840 Lamé and Lebesgue gave proofs for n = 7. The proposition appears to be true universally, and in 1849 Kummer, by means of ideal primes, proved it to be so for all numbers except those (if any) which satisfy three conditions. It is not certain whether any number can be found to satisfy these conditions, but there is no number less than 100 which does so. The proof is complicated and difficult, and there can be no doubt is based on considerations unknown to Fermat. I may add that, to prove the truth of the proposition, when n is greater than 4 obviously it is sufficient to confine ourselves to cases when n is a prime, and the first step in Kummer's demonstration is to shew that one of the numbers x, y, z must be divisible by n. The following extracts, from a letter now in the university library at Leyden, will give an idea of Fermat's methods; the letter is undated, but it would appear that, at the time Fermat wrote it, he had proved the proposition (h) above only for the case when n = 3. Je ne m'en servis au commencement qe pour demontrer les propositions negatives, comme par exemple, qu'il n'y a aucu nombre moindre de l'unité qu'un multiple de 3 qui soit composé d'un quarré et du triple d'un autre quarré. Qu'il n'y a aucun triangle rectangle de nombres dont l'aire soit un nombre quarré. La preuve se fait par apagogeen en cette manière. S'il y auoit aucun triangle rectangle en nombres entiers, qui eust son aire esgale à un quarré, il y auroit un autre triangle moindre que celuy la qui auroit la mesme proprieté. S'il y en auoit un second moindre que le premier qui eust la mesme proprieté il y en auroit par un pareil raisonnement un troisieme moindre que ce second qui auroit la mesme proprieté et enfin un quatrieme, un cinquieme etc. a l'infini en descendant. Or est il qu'estant donné un nombre il n'y en a point infinis en descendant moindres que celuy la, j'entens parler tousjours des nombres entiers. D'ou on conclud qu'il est donc impossible qu'il y ait aucun triangle rectange dont l'aire soit quarré. Vide foliu post sequens.... Je fus longtemps sans pouvour appliquer ma methode aux questions affirmatives, parce que le tour et le biais pour y venir est beaucoup plus malaisé que celuy dont je me sers aux negatives. De sorte que lors qu'il me falut demonstrer que tout nombre premier qui surpasse de l'unité un multiple de 4, est composé de deux quarrez je me treuvay en belle peine. Mais enfin une meditation diverses fois reiterée me donna les lumieres qui me manquoient. Et les questions affirmatives passerent par ma methods a l'ayde de quelques nouveaux principes qu'il y fallust joindre par necessité. Ce progres de mon raisonnement en ces questions affirmatives estoit tel. Si un nombre premier pris a discretion qui surpasse de l'unité un multiple de 4 n'est point composé de deux quarrez il y aura un nombre premier de mesme nature moindre que le donné; et ensuite un troisieme encore moindre, etc. en descendant a l'infini jusques a ce que vous arriviez au nombre 5, qui est le moindre de tous ceux de cette nature, lequel il s'en suivroit n'estre pas composé de deux quarrez, ce qu'il est pourtant d'ou on doit inferer par la deduction a l'impossible que tous ceux de cette nature sont par consequent composez de 2 quarrez. Il y a infinies questions de cette espece. Mais il y en a quelques autres que demandent de nouveaux principes pour y appliquer la descente, et la recherche en est quelques fois si mal aisée, qu'on n'y peut venir qu'avec une peine extreme. Telle est la question suivante que Bachet sur Diophante avoüe n'avoir jamais peu demonstrer, sur le suject de laquelle Mr. Descartes fait dans une de ses lettres la mesme declaration, jusques la qu'il confesse qu'il la juge si difficile, qu'il ne voit point de voye pour la resoudre. Tout nombre est quarré, ou composé de deux, de trois, ou de quatre quarrez. Je l'ay enfin rangée sous ma methode et je demonstre que si un nombre donné n'estoit point de cette nature il y en auroit un moindre que ne le seroit par non plus, puis un troisieme moindre que le second etc. a l'infini, d'ou l'on infere que tous les nombres sont de cette nature.... J'ay ensuit consideré questions que bien que negatives ne restent pas de recevoir tres-grande difficulté, la methods pour y pratiquer la descente estant tout a fait diverse des precedentes comme il sera aisé d'espouver. Telles sont les suivantes. Il n'y a aucun cube divisible en deux cubes. Il n'y a qu'un seul quarré en entiers que augmenté du binaire fasse un cube, ledit quarré est 25. Il n'y a que deux quarrez en entiers lesquels augmentés de 4 fassent cube, lesdits quarrez sont 4 et 121.... Apres avoir couru toutes ces questions la plupart de diverses (sic) nature et de differente façon de demonstrer, j'ay passé a l'invention des regles generales pour resoudre les equations simples et doubles de Diophante. On propose par exemple 2 quarr. + 7957 esgaux a un quarré (hoc est 2xx + 7967 \propto quadr.) J'ay une regle generale pour resoudre cette equation si elle est possible, on decouvrir son impossibilité. Et ainsi en tous les cas et en tous nombres tant des quarrez que des unitez. On propose cette equation double 2x + 3 et 3x + 5 esgaux chaucon a un quarré. Bachet se glorifie en ses commentaires sur Diophante d'avoir trouvé une regle en deux cas particuliers. Je me donne generale en toute sorte de cas. Et determine par regle si elle est possible ou non.... Voila sommairement le conte de mes recherches sur le sujet des nombres. Je ne l'ay escrit que parce que j'apprehende que le loisir d'estendre et de mettre au long toutes ces demonstrations et ces methodes me manquera. En tout cas cette indication seruira aux sçauants pour trouver d'eux mesmes ce que je n'estens point, principlement si Mr. de Carcaui et Frenicle leur font part de quelques demonstrations par la descente que je leur ay envoyees sur le suject de quelques propositions negatives. Et peut estre la posterité me scaure gré de luy avoir fait connoistre que les anciens n'ont pas tout sceu, et cette relation pourra passer dans l'esprit de ceux qui viendront apres moy pour traditio lampadis ad filios, comme parle le grand Chancelier d'Angleterre, suivant le sentiment et la devise duquel j'adjousteray, multi pertransibunt et augebitur scientia. (ii) I next proceed to mention Fermat's use in geometry of analysis and of infinitesimals. It would seem from his correspondence that he had thought out the principles of analytical geometry for himself before reading Descartes's Géométrie, and had realised that from the equation, or, as he calls it, the ``specific property,'' of a curve all its properties could be deduced. His extant papers on geometry deal, however, mainly with the application of infinitesimals to the determination of the tangents to curves, to the quadrature of curves, and to questions of maxima and minima; probably these papers are a revision of his original manuscripts (which he destroyed), and were written about 1663, but there is no doubt that he was in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629. He obtained the subtangent to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e; but there is nothing to indicate that he was aware that the process was general, it is probable that he never separated it, so to speak, from the symbols of the particular problem he was considering. The first definite statement of the method was due to Barrow, and was published in 1669. Fermat also obtained the areas of parabolas and hyperbolas of any order, and determined the centres of mass of a few simple laminae and of a paraboloid of revolution. As an example of his method of solving these questions I will quote his solution of the problem to find the area between the parabola y³ = p x², the axis of x, and the line x = a. He says that, if the several ordinates of the points for which x is equal to a, a(1 - e), a(1 - e)²,... be drawn, then the area will be split into a number of little rectangles whose areas are respectively ae(pa^2)^(1/3), ae(1-e) ( pa^2(1-e)^2 )^(1/3),... . The sum of these is p^(1/3) a^(5/3) e / ( 1 - (1 - e)^(5/3) ) ; and by a subsidiary proposition (for he was not acquainted with the binomial theorem) he finds the limit of this, when e vanishes, to be (3/5) p^(1/3) a^(5/3) . The theorems last mentioned were published only after his death; and probably they were not written till after he had read the works of Cavalieri and Wallis. Kepler had remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this principle to a few examples. Thus, to find the maximum value of x(a - x), his method is essentially equivalent to taking a consecutive value of x, namely x - e where e is very small, and putting x(a - x) = (x - e)(a - x + e). Simplifying, and ultimately putting e = 0, we get x = œ. This value of x makes the given expression a maximum. (iii) Fermat must share with Pascal the honour of having founded the theory of probabilities. I have already mentioned the problem proposed to Pascal, and which he communicated to Fermat, and have there given Pascal's solution. Fermat's solution depends on the theory of combinations, and will be sufficiently illustrated by the following example, the substance of which is taken from a letter dated August 24, 1654, which occurs in the correspondence with Pascal. Fermat discusses the case of two players, A and B, where A wants two points to win and B three points. Then the game will be certainly decided in the course of four trials. Take the letters a and b, and write down all the combinations that can be formed of four letters. These combinations are 16 in number, namely, aaaa, aaab, aaba, aabb; abaa, abab, abba, abbb; baaa, baab, baba, babb; bbaa, bbab, bbba, bbbb. Now every combination in which a occurs twice or oftener represents a case favourable to A, and every combination in which b occurs three times or oftener represents a case favourable to B. Thus, on counting them, it will be found that there are 11 cases favourable to A, and 5 cases favourable to B; and since these cases are all equally likely, A's chance of winning the game is to B's chance as 11 is to 5. The only other problem on this subject which, as far as I know, attracted the attention of Fermat was also proposed to him by Pascal, and was as follows. A person undertakes to throw a six with a die in eight throws; supposing him to have made three throws without success, what portion of the stake should he be allowed to take on condition of giving up his fourth throw? Fermat's reasoning is as follows. The chance of success is 1/6, so that he should be allowed to take 1/6 of the stake on condition of giving up his throw. But if we wish to estimate the value of the fourth throw before any throw is made, then the first throw is worth 1/6 of the stake; the second is worth 1/6 of what remains, that is 5/36 of the stake; the third throw is worth 1/6 of what now remains, that is, 25/216 of the stake; the fourth throw is worth 1/6 of what now remains, that is, 125/1296 of the stake. Fermat does not seem to have carried the matter much further, but his correspondence with Pascal shows that his views on the fundamental principles of the subject were accurate: those of Pascal were not altogether correct. Fermat's reputation is quite unique in the history of science. The problems on numbers which he had proposed long defied all efforts to solve them, and many of them yielded only to the skill of Euler. One still remains unsolved. This extraordinary achievement has overshadowed his other work, but in fact it is all of the highest order of excellence, and we can only regret that he thought fit to write so little. _________________________________________________________________ This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908). Transcribed by D.R. Wilkins (dwilkins(a)maths.tcd.ie) School of Mathematics Trinity College, Dublin

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Re: rant on the morality of confidentiality (fwd)
by Jim Choate 17 Dec '03

17 Dec '03

Forwarded message: > Date: Sat, 10 Jan 1998 16:51:05 -0800 > From: Tim May <tcmay(a)got.net> > Subject: Re: rant on the morality of confidentiality > Doesn't matter how the establishment (whatever that might be) looked on him > or not...you challenged me to name _one_ example, and I named several. Oh, > and it is not true as you later claim that all of my examples "eventually > published" all of their findings. Fermat did not, Gauss did not. Gauss and Fermat both published or shared their works with others contemporaries as demonstrated via their short biographies that I submitted to the list. > My main point has been to refute your notion that any one who elects not to > publish in the open literature cannot be a scientist. I know of many > scientists who could not publish, or chose not to for various reasons. If one doesn't publish (which in the scientific sense means to share publicly ones discoveries, *not* necessarily in open literature - which is a constraint you have never implied before now - changing the rules in the middle of the game are we...and while you go ballistic - when Gauss and Fermat were alive there were no magazines, work was published via demonstrations, books or by letters shared between co-workers) then be so kind as to explain how anyone would find out about the work in the first place, divine intervention? > I mentioned the Manhattan Project scientists. (Choate made some bizarre > claim after this mention that all of the science was known in the 20 and > 30s, and that no actual science was done by MP "engineers" and > "technicians." Might be a surprise to Ulam, Teller, von Neumann, and all > the others who worked in secrecy on the atom bomb, then the hydrogen bomb, > and so on.) What Ulam, Teller, Von Neumann did regarding actualy making the devices; such as the work on explosive lenses; was *engineering or technology* it wasn't science. Science as was extant at the time was quite capable of describing what needed to be done, the question was how to build the damn thing. THAT question is technological not scientific. Furher, at *no* time did I imply that those working on the bomb simply quite working on the theoretical issues that were still unanswered. The reality is, as my original claim, that the process *required both*. However, taking this to the extreme, as you are want to do, of equating them is a disservice to their achievements and the endeavors of science and engineering. Further, if you had done one whit of research on the Manhatten Project you would have found that the *primary* issue for most of the scientist was that the military wouldn't allow them to *share their work*. In many of the letters, depositions, minutes of meetings, etc. there are continous and heated discussion on how the secrecy impacted negatively the process of meeting their goals. Ulam, Teller, Von Neumann, et ali. would be the *first* to refute your claim. > Oh, and what of all the many fine Russian scientists of this century, > nearly all restricted in what they could publish? Because they could not > submit their work to open publication were they not doing science? And very little of what they didn't publish outside of the CCCP made a big difference to what others were doing. As a matter of fact, because of this many of the Russian scientists are reaping the rewards by being passed over for Nobels and other such rewards for sharing their work. A very clear result of this policy in the CCCP was the technological 2nd class that resulted in just about every aspect of Soviet science and engineering outside of some very esoteric and theoretical work. A primary example of *why* and *how* such secrecy *inhibits* rather than promotes science is a study of Russian biological sciences. If you seriously claim that the policy of state secrecy didn't inhibit russian science then be so kind as to explain their clones of the Apple II or the IBM 360, which were truly attrocious in their technology. Just look at the impact on China because of these sorts of policies, it was in the late 1970's before they ever managed to build a TTL equivalent quad-NAND gate, something that the western world had been cranking out since the late 1960's. I got involved in computers and electronics in 1969 *because* those self same 7400 chips were *dumped on the surplus market*. Prey tell how we managed to create a technological millieu that allows 9 year old kids to play with chips for pennies when a whole damn country with a population in the billions can't manage to build and use till nearly 10 years later? > The point being that open publication is only a part of the methodology of > doing science, and a fairly recent one, too. Malarky. If you simply review the history of science (something that is becoming clear you haven't) and study some of these great luminaries works you find that *critical* to all of them was *sharing* their work(s) with others of a similar bent. Their biographies are replete with mentions of letters, meetings, books, etc. they both supplied and received from others. Back to Newton, he graduated in 1665 w/ a BA from Cambridge. Shortly before this he had begun his experiments in light. He published these results in 'Philosophical Transactions' in 1672 and was elected a fellow of the Royal Society the same year because of it. In 1687, at Halleys prompting to abandon his chemical studies (he was an alchemist) and focus on mechanics, he published 'Principia'. This same year is when he became involved in politics which eventualy resulted in his appointment in 1689 to the Convention Parliament of the university. In 1695 he was appointed the Warden of the Mint. In 1727 he was appointed Master of the Mint, which he held until his death. In 1703 he became the president of the Royal Society. That same year he published 'Opticks'. In 1705 he was knighted. He also wrote religous works, two of the best known were 'Observations of the Prophecies of Daniel' and 'Church History'. How in good faith you can claim this doesn't qualify as sharing their work and that such sharing isn't critical to advancement is truly revealing. ____________________________________________________________________ | | | Those who make peaceful revolution impossible will make | | violent revolution inevitable. | | | | John F. Kennedy | | | | | | _____ The Armadillo Group | | ,::////;::-. Austin, Tx. USA | | /:'///// ``::>/|/ http://www.ssz.com/ | | .', |||| `/( e\ | | -====~~mm-'`-```-mm --'- Jim Choate | | ravage(a)ssz.com | | 512-451-7087 | |____________________________________________________________________|

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More on Fermat...
by Jim Choate 17 Dec '03

17 Dec '03

Hi, Given Tim May's assertion that publishing is not critical to the advancement of science *and* that Fermat is a prime example of this I would like to examine it. Let's ignore the references to Fermat's letters and such to Pascal, Descarte, and others. Let's take as a given that Fermat's work was not published. When he died the executors found the material and decided to publish it (we'll ignore their motives for a moment). The result was a global 'aha' for mathematicians. A collective "Ah, so THAT's how you do that." Since it is clear that had Fermat's work *not* been published those self same problems would have remained unsolved, perhaps some as long as his Last Theorem which stands as a prime example of the result of *not* publishing or noting results, perhaps some even till today. It is clear that the initial hypothesis that publishing (or sharing) of work is not critical is clearly incorrect by the very example it holds to prove itself. ____________________________________________________________________ | | | Those who make peaceful revolution impossible will make | | violent revolution inevitable. | | | | John F. Kennedy | | | | | | _____ The Armadillo Group | | ,::////;::-. Austin, Tx. USA | | /:'///// ``::>/|/ http://www.ssz.com/ | | .', |||| `/( e\ | | -====~~mm-'`-```-mm --'- Jim Choate | | ravage(a)ssz.com | | 512-451-7087 | |____________________________________________________________________|

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area 51, enviro crime, secrecy==abuse?
by Vladimir Z. Nuri 17 Dec '03

17 Dec '03

Environment of Secrecy A lawsuit alleges environmental crimes at the country's most secret military base Amicus Journal, Spring 1997, a publication of the National Resources Defense Council by Malcom Howard August, 1994: Standing atop a desert ridge in central Nevada, Glenn Campbell peers through binoculars at a remote duster of buildings in the valley below. "It's the most famous secret militaryfacility in the world," he says. The scattering of airplane hangars and radar dishes below, barely visible through the haze, is a secluded Air Force test facility known as Area 51-- or, more fancifully, "Dreamland"-- that is believed to have launched the most sophisticated Cold War aircraft, from supersonic spy planes to the radar-evading Stealth bomber. Campbell has made a mini-industry of showing off this clandestine outpost, built on a barren pancake of alkali just inside the Air Force's restricted Nellis Gunnery Range north of Las Vegas. His self-published tour book describes how to get a stealthy, yet fully legal, view of Area 51. Tourists pass electronic sensors on the road and watch helicopters patrol above, and are tailed by men in unmarked white jeeps who train high-powered video cameras on their every move. Though Campbell's tour stays entirely on public land, once on the ridge his clients stand only yards from the Area 51 boundary. Signs prohibiting photography and warn that "use of deadlyforce" is authorized against trespassers. These days, Glenn Campbell's not-for-profit tour business has fallen on hard times. In 1995, the Air Force all but shut him down: it seized the 4,500 acres of public land where Campbell's customers used to get their best views. The move demonstrates just how touchy the Air Force is about this military sanctum sanctorum-- since, in order to close out a few ragtag sightseers, it inevitably whipped up a storm of speculation among the conspiracy buffs, tabloid press, UFO trackers, aviation hounds, and government accountability activists who are fascinated by Dreamland. One can only imagine, then, the consternation in the upper ranks of the Air Force when four former Area 51 employees and widows of two others brought their now celebrated lawsuit, alleging that the secrecy surrounding the site had been used to commit and then cover up environmental crimes. "My husband came home one day screaming," says Helen Frost, whose late husband, Robert, was a sheet- metal worker at Area 51. "He was screaming, 'My face is on fire.' His face was bright red and swollen up like a basketball. Then he got three- inch scars on his back. A year later, he died." In 1990, the year after Frost's death, a posthumous worker's compensation hearing found that the liver disease that killed him stemmed from heavy drinking, not toxic industrial chemicals. But Helen Frost disputes that finding. She points to testimony from a Rutgers University chemist who found high levels of dioxins and dibenzofurans in her husband's tissue. Those extremely dangerous chemicals, wrote Dr. Peter Kahn-- best known for his role in the Agent Orange commission-- were likely the result of industrial exposure. Helen Frost and her co-plaintiffs filed the original lawsuit in 1994, alleging that the military and its contractors regularly and illegally burned huge volumes of toxic waste in the desert, exposing workers to dangerous fumes. Defense contractors from the Los Angeles area, they claimed, routinely trucked 55-gallon drums full of paints and solvents into Area 51. Employees would dig large trenches, toss in the drums, spray on jet fuel, and finally light the toxic soup with a flare. The plaintiffs named the Department of Defense, the National Security Agency, and the Air Force in the suit, charging that they allowed the burning in violation of the Resource Conservation and Recovery Act (RCRA), the nation's keystone hazardous waste law. In a parallel suit, they charged the Environmental Protection Agency (EPA) with failing to inspect and monitor waste disposal at the facility, as RCRA requires. The plaintiffs have said that many other Area 51 workers are suffering from ailments similar to Frost's. They do not seek damages-- just information about what chemicals they were exposed to, help with their medical bills, and an end to the burning. The extreme secrecy shrouding Area 51 has turned the lawsuit into something out of a Cold War spy novel, replete with sealed motions, confidential hearings, blacked-out docket sheets, and classified briefings. "We're in the rather unenviable position of suing a facility that doesn't exist, on behalf of workers who don't officially exist," says Jonathan Turley, the George Washington University law professor who is representing the plaintiffs. The existence of the workers is fairly straightforward: because they took secrecy oaths in order to work at Dreamland, they fear recrimination for going to court, and so the judge has allowed them to sue anonymously. But the existence of Area 51 is more problematic. The base is absent from even the most detailed defense flight charts. Ask the Air Force communications office about the facility, and a spokesman will read from a script: "There is an operating location in the vicinity of Groom Dry Lake. Some specific activities conducted on the Nellis Range both past and present remain classified and can't be discussed." In court, the Air Force tactics have been just as convoluted. In the early days of the lawsuit, argued before U.S. District Court Judge Phillip M. Pro, much of the contention centered on the Air Force's refusal to name the place at issue. The plaintiffs have all sworn that they worked at a facility called "Area 51," and Turley has introduced evidence, such as his clients' employee-evaluation forms and various government documents, that refer to the site as "Area 51." Air Force lawyers, however, have said that naming the base would undermine national security, because enemy powers could make valuable inferences from any verified names. In response, the plaintiffs accused the Air Force of cynically invoking national security in order to wriggle out from under the evidence that illegal practices were going on at a place called "Area 51." After all, Turley argued in court, "If the defendants confirmed 'Area 51' is often used to identify this facility, a foreign power would be no more educated as to [the facility's] operations than their previous knowledge, derived in no small part by the defendants' own public statements." But the name of the facility was only the first of a barrage of secrecy arguments the plaintiffs have faced. Throughout pretrial proceedings, Air Force lawyers repeatedly invoked the military and state secrets privilege, a rarely used tenet of common law that allows the executive branch to withhold information from trial if its disclosure might jeopardize U.S. soldiers or diplomatic relations. To support the claim, Air Force Secretary Sheila Widnall submitted two afffidavits, one public and one for the judge's eyes only, in which she argued that any environmental review of the facility entered into the record could educate foreign powers about U.S. military technology. "Collection of information regarding air, water, and soil is a classic foreign intelligence practice because analysis of these samples can result in the identification of military operations and capabilities," Widnall wrote. Turley-- himself a former staff member of the National Security Agency-- believes that the Air Force is improperly using the military secrets privilege to hamstring his case. Most of the chemicals burned at Area 51, he says, were standard solvents, paints, and the Like that are found at any aircraft production facility. If sensitive data did emerge, such as traces of the chemicals used in the radar-blunting coat of the Stealth fighter, they could simply be stricken from the record. Whatever the case, so far the tactics of the Air Force have largely prevailed. True, the plaintiffs have changed the course of environmental policy at the base; because of their suit, the Justice Department has launched a criminal investigation into the charges on EPA's behalf, and EPA has conducted the first hazardous waste inventory of Area 51. But that inventory remains off limits to the plaintiffs, even though RCRA requires EPA to make such documents public because Judge Pro ruled that the president could grant a special exemption for national security reasons. RCRA has always allowed a president to create this kind of exemption; what is unusual about this case is that the judge allowed a president to do so after allegations of environmental crime had already emerged. And the exemption was duly granted: late in 1995, President Clinton signed an executive order exempting Area 51 "from any Federal, State, interstate or local provision respecting ... hazardous waste disposal that would require the disclosure of classified information ... to any unauthorized person." In the wake of the president's intervention, in the spring of 1996 Judge Pro dismissed the main case against the Pentagon on national security grounds. Turley has appealed the ruling to the Ninth Circuit Court of Appeals. To date, the court has not issued a ruling. In some senses, the lawsuit is unique: there is only one Area 51. The military has dozens of other restricted bases where highly secret weapons tests are carried out-- but, to the best of any civilian's knowledge, all of these sites are already listed on EPA's dockets. Environmental information about standard military bases is freely available. In general, says NRDC nuclear arms expert Stan Norris, the Air Force's behavior in the Area 51 case is "not representative of the Department of Defense. They're not naturally secretive in [the environmental] area." Compared to the environmental traditions of the Department of Energy-- which opened up information on its nuclear weapons production sites only after years of public pressure and lawsuits-- when it comes to the Department, Norris says, "We're awash in information." But Turley and other students of military secrecy believe that at issue in the Area 51 case is a bedrock principle. "In the end, this case can be boiled down to one question," says Turley "Can the Department of Defense create secret enclaves that are essentially removed from all civilian laws and responsibilities?" Borrowed from English common law, the military and state secrets privilege is as old as the nation itself. Ever since Aaron Burr stood trial for treason in 1807, the executive branch has, from time to time, sought to block information in civil and criminal trials. In Burr's case, the government refused to release letters written by one of Thomas Jefferson's generals. The defendant swore the letters would clear his name, but federal lawyers argued that the private notes "might contain state secrets, which could not be divulged without endangering the national safety." The secrecy powers were used most heavily during the Cold War, when military and intelligence agencies sought to hide technology from the Soviets and protect eavesdropping methods used against civilian activists. The Dreamland litigation, however, marks the first time the military and state secrets privilege has been invoked in a civil suit over toxic waste. It represents a fundamental clash between the demands of national security, in which stealth is an asset, and the right of public scrutiny that is at the core of U.S. environmental laws. National security and environmental law scholars take a keen interest in the case. "It seems to me that specific details of weapons programs can properly be held secret," comments Kate Martin, director of the Center for National Security Studies, which litigated some of the key state secrets cases of the 1980s. "The question is, is secrecy being used as a way of of avoiding accountability, compliance with environmental law, or worker-safety standards?" Others see such speculation as both paranoid and naive. "Just because the Soviet Union is no longer around doesn't mean we don't need to keep secrets," says Kathleen Buck, former Pentagon general counsel for President Reagan. She argues that, since President Clinton's defense review revealed continued threats of ballistic missile attack, nuclear proliferation, rogue states, and terrorist cells, secrecy is a strategic advantage the United States still needs. "But we have to make sure that in building up the national defense, we don't destroy the very thing we're trying to protect," objects Steve Dycus, professor of national security and environmental law at Vermont Law School. A victory for the Pentagon over Area 51, he believes, could frustrate EPA's efforts to enforce environmental laws at sensitive military sites-- and the Pentagon, with more than a hundred active Superfund sites, is considered by many to be among America's worst polluters. Moreover, a military victory could have a chilling effect on other military employees who find themselves considering the difficult act of whistleblowing. After all, Dycus notes, RCRA is designed in part to enlist the help of citizens and states in enforcing environmental protection. While scholars debate policy, the employees of Area 51 wait for justice. The Air Force denies the charge of illegal burning, and Judge Pro dismissed the lawsuit without deciding on its substantive charges; so the plaintiffs have no answers to their questions about the painful skin disorders they say they suffer from. And, unless their appeal to the Ninth Circuit is granted, President Clinton's exemption precludes them from obtaining any information about what they might have been exposed to. Ironically, that exemption was made public the same day Clinton announced that the government would compensate victims of nuclear radiation experiments. "Our greatness is measured not only in how we so frequently do right," he said, "but also how we act when we have done the wrong thing." Has the United States done the wrong thing at Area 51? Without some kind of break in the intense secrecy that surrounds the place, the public has no way of knowing. To Glenn Campbell, who has made it his life's work to inform Americans about Area 51, the existence of this level of concealment-- and the lack of accountability that comes with it-- are cause for suspicion. "The military is the only governmental branch that has the prerogative to keep things secret from the public," he says. "The problem is, where there's excessive secrecy, there's usually abuse."

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Re: rant on the morality of confidentiality
by Blanc 17 Dec '03

17 Dec '03

Vladerusky: Your original post brought up several separate aspects which can be considered separately and may not necessarily coexist in the same place at the same time: 1) secrecy 2) responsibility for publishing 3) working for the government at the expense of unwilling payors 4) the motivations of "true scientists" 5) the requirements for the advancement of science 6) the need of science for the works of great minds Initially your argument had to do with secrecy and the need for scientists to publish their work so that the scientific community may benefit from it. I can't disagree that if a scientist is working for the public, that they should make their work publically available to them, since, after all, they are supposedly working for the public benefit. But you also said that "a key aspect of SCIENCE is publishing". I was only pointing out that, in the context of those who are working for their own purposes and not under the employment of a government agency, some scientists are not overly concerned about contributing to this advancement, as can be observed by their reluctance to publish (even if they eventually do, "under the extreme pressure of friends", for instance). It may be your conclusion that the advancement of science depends upon scientists publishing their works, but the fact is that some great scientists, and many others as well, are not as motivated to contribute as you think is proper for a "true scientist". I think you should distinguish between those scientistis who have joined some kind of "scientific community" and have established an obligation to share the results of their work with that group, and those scientists who are what they are, and do what they do, from motivations unrelated to such communities. .. Blanc

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