CA2550259A1 - Verifiable, secret shuffles of encrypted data, such as elgamal encrypted data for secure multi-authority elections - Google Patents

Abstract

A cryptographic process permits one to verifiably shuffle a series of input data elements. One or more authorities or individuals "shuffle", or "anonymize" the input data (e.g. public keys in discrete log form or ElGamal encrypted ballot data). The process includes a validity construction that prevents any one or more of the authorities or individuals from making any changes to the original data without being discovered by anyone auditing a resulting proof transcript. The shuffling may be performed at various times.
In the election example, the shuffling may be performed, e.g., after ballots are collected or during the registration, or ballot request phase of the election, thereby anonymizing the identities of the voters.

Description

VERIFIABLE, SECRET SHUFFLES OF ENCRYPTED DATA, SUCH
AS ELGAMAL ENCRYPTED DATA FOR SECURE
MULTI-AUTHORITY ELECTIONS
TECHNICAL, FIELD
The following relates generally to encryption, and more specifically to electronic encryption such as for use in voting schemes.
BACKGROUND AND SL~I~IARY
The notion of a shuffle of a collection of objects, records, or tokens is simple and intuitive, and useful examples abound in various daily human activities. A gambler in a casino knows that when he picks up his hand of cards, each one will be one of 52 unique values, and that no one else at the table will have duplicates of the ones he holds. He does not, however, have any knowledge of how the cards are distributed, even though he may have recorded the exact card order before they were shuffled by the dealer.
In the context of electronic data, the problem of achieving the same kind of random, yet verifiable pe~nutation of an input sequence is surprisingly difficult. The problem is that the data itself is either WO 01/'7369a PC'IYLTS0r109550 always visible to the auditor, or it isn't. If it is, then the comespondeace between input records end output records is trivial to reconstruct by the auditor or other observer. If it isn't, then input and output records must be different representations of the same underlying data. Bnt if the output is different enough (that is, encrypted well enough) that the auditor cannot reconstruct the correspondence, then how can the auditor be score that the shu$Ier did not change the underlying data in the process of shuffling?
Most of the description below is devoted to giving an e~cient (linear) method for solving this problem in an important context - ElGamal encrypted data. In order to make the exposition as clear and concise as possible, the majority of the description below explicitly refers to the specific case when operations are carried out in Zl, the multiplicative group of units modulo a large prime, p. However, the only properties of the underlying (multiplicative) group used is that the associated ElGamal cryptosystem should be secure, and that the Chaum-Pedersen protocol for the relation logs X = loge, Y = a (D. Chaum_ Zero-knowledge undeniable signatures. Advances is Cryptology -EUROCRYPT '90, Lecture Notes is Computer Science, volume 473, pages 458-464, Springer-Verlag, 1991. D. Chaurn and T.P. Pederserr.
Wallet Daxabases With Observers. In Advance~c in Cryptologyr --CRYP~'O '91, Volume 740 of Lecture Notes in Computer Scfence, pages 89-105, Berlin, 1993. Springer-Verlag.) should not leak information about floe secret exponent, r<. 1n fact, for one embodiment, a universally verifiable, mufti authority election protocol - the verifier will be implemented via the Fiat-Shatnir heuristic (A. Fiat, A. Shamir. How to --prove yourself-. Practical solutions to identification and signature problems. Adv~ces in Cryptology - CRYPTO '86, Lecture Notes in Computer Science, pp. 186-194, Springer-Verlag, New York, 1987,), so in this case it is sufficient that the protocol be zero-knowledge against an honest verifier. (R. Cramer, R. Geanaro, B. Schoenmakers. A secure and optimally efficient mufti-authority election scheme. Advances in Cryptology - EUROCRYPT '97, Lecture Notes in Computer Science, SUBSTITUTE SHEET (RULE 26) WO O1h369.i PCTNSO1/09550 Springer-Verlag, 1997.) Thus, ~c shuffle protocol is also useful when the ElGamal cryptosystem is implemented over other groups such as elliptic curves. 'the general Boolean proof techniques of R Cramer, I.
Damgrd, B. Schoenmakers, Proofs of partial k~wwledge and simplified design of witness hiding protocols (Advances in Clyptology--~RYPTO
'94, Lecture Notes is Computer Science, pp. 174-187, Springer-Verlag, Berlin, 1994.), can also be used to construct a proof with the same properties, however, the resulting proof size (complexity) is quadratic,. or worse, in the size of the input sequence.
The protocols or methods described below also offer several advantages over the cut-and-choose technique as used in K. Sako, J.
Kilian. Receipt $ce mac-type voting scheme-A practical solution to the implementation of a voting booth, Advances in Cryptology-EUROCRYP? '95, Lecture Notes in Computer Science, Spzinger-Verlag, 1995. In this approach, the siu of the proof is dependent on the probability of a cheatit~ proven that is required to satisfy all participants.
In the shuffle protocol described herein, this cheating probability is essentially k/q, where k is the number of elements to be shuffled, and q is the siau of the subgroup of Zl in which ,the elements are encrypted.
Although no analysis of the proof size is done in the K. Sako paper, it $ppesrs that, in order to obtain similarly low cheating probability, ii will need to be orders of magnitude larger than the siu of the proof provided herein. (Moreover, if the K. Sako protocol is implemented non-interactively, the cheating probability would need to be chosen exceedingly small, because a malicious participant might use considerable ofF-l~iune computation to generate a forged proof by exhaustive search_ This of course, could he the case with the protocols described, but the probability k/g is, for all practical values of k and q, certainly small enough.) The advantage of the current scheme becomes even more apparent when sew in the context of the resulting universally verifiable election protocol. In K Sako, each voter must interact sequentially with SUBSTITUTE SHEET (RULE 26) we oin3s~ rcrnrsoiro9sso each "counting center" before actually casting his/her vote. On this account; it is unlikely that a useable implementation could be built for large scale, public sector elections in the near future. In contrast, protocols described below, put all authority participation (except, possibly, for the creation of basic election parameters) at the close of the election, purely for the purpose of tabulation.
This nice property is also found in the elegant homomorphic election protocol in the paper by R Cramer, R Gennaro, and B.
Schoenmskers. However, that protocol can only be applied to ballots whose questions are of a simple "choose (at most) m of n" type. Ibis effectively precludes "wsite-in" responses, as well as "proponional type"
questions where the voter is expected to indicate answers in preferential order, and questions are tabulated in accordance with this preference. A
couple of somewhat less_importaat disadvantages of the R Cramer, R
GGnnaro, and B. Sehoenmakers scheme are that it expands vote data siu considerably, and that it requires a voter validity proof. This proof further expands the vote data size by about an order of magnitude, and is unattractive from a practical perspective, because it presumes special purpose code to he running on the voter's computer.
The protocol is described below constructed entirely from a sequence of Chaum-Pedersen proofs, and elementary arithmetic operations. It is thus simple to implement.
BRIEF DESCRIPTION OF THE DRAWIrTC3~S
Figure 1 is a block diagram illustrating a suitable w environment for implemactting embodiments of the invention.
Figure 2 is a schematic diagram illustrating a simple implementation of the shu$le protocol descn'bed herein as applied to a simple ballot with three voters and three shuttles.
Figwe 3 is a flow diagram illustrating a scaled iterative logarithmic multiplication proof protocol.

SUBSTITUTE SHEET (RULE 26) WO o117369s ~CT/USO1/09550 Figure 4 is a flow diagram. illustrating a simple shuffle protocol where the shuf~ler knows the encryption exponent_ Figure 5 is a flow disgiam illustrating a general shuffle protocol where the shu$ler does not lmow the encryption exponents.
Figure 6 is a flow diagram illustrating an anonymous certificate distribution routine.
Figure 7 is a flow diagram illustrating an alternative embodiment to the anonymous certificate distribution routine of Figure 6.
Ia the drawings, identical reference numbers identify identical or substaatislly similar elements or acts. To easily identify the discussion of any particular element or act, the most significant digit or digits is a reference number refer to the Figure number itt which that element is first introduced (e.g., element X04 is first introduced and discussed with respect to Figure 2).
The headings provided herein are for convenience only and do not necessarily a~'ect the scope or meaning of the daimed invention.
DETAILED DESCRIPTION
1. Overveew Described in detail below is a cryptographic protocol to vesihably ah~le a series of eletaeats, :uch as an input sequence of public keys in discrete log form or k input ElGamal encryptions, that has applications ill, e.g., a secure, universally verifiable, mufti-authority election scheme. In the case of k input ElGamal encryptions, the output of the shuffle operation is another sequence of k ElGamal encryptions, tech of which is a re-rncryption (i.e., as ElGamal pair which encrypts exactly the same clear text) of exactly one of the input encryptions, but the order of elements, in the output is kept secret. Though it is a trivial matter for the "shu$ler" (who chooses the permutation of the elements to be applied and the encryption keys used) to compute the output from the input, the construction is important because it provides a linear size proof SU95TtTUTE SHEET (RULE 26) WO 01/~369d pCT/Ugpl/09550 of coorrccmess f~ the output sequence (r. e_, a proof that it is of the form claimed) that can be cltecaced by an arbitrary verifier. The security of the proof protocol is the same as that of the Chaum Pedersen protocol for the relation logs~r = logs, y, which is sufTcient for the election application in which it as used.
The following description provides specific details for a thorough understanding o~ and enabling description for, embodiments of the invention. However, one skilled in the art will understand that the invention may be practiced without these details. In other instances, well laaown structtars and functions have not been shown or described in detail to avoid unnecessarily obscuring the description of the embodiments of the invention.
Much of the detailed description provided below is explicitly disclosed is tl~,e provisional patent applications noted above;
much of the additional material will be reco~nized by those skilled in the relevarst art ss being inherent is the detailed description provided in such provisional patent applications, or well known to those skilled in the relevant art. Those skilled is the relevant art can implement aspects of the invention based on the detailed description provided in the provisional patent applications.
The mathematical aotatian used here is readily understandable to those skilled in the relevant art; however, for those unfamiliar with the art, the following definitioons and descriptions are provided. Such definitions, although brief, will help those generally unfamiliar with the art to more fully understand aspects of the invention .based on the detailed description provided herein. Such definitions are further defined by the description of the invention as a whole (including the claims), and not simply by such definitions.
Figures 1-5, as well as the detailed description provided herein, describe protocols betaroea a party (e.g., a vatzr) and a verifier (or between a proving Patty and a vorifyi~ng Party). The actions performed by the parties are grouped together into flow diagram boxes. Tn general, each SUBSTITUTE SHEET (RULE 26) WO O1I13G94 PCTIITSoI/09550 line of text or equations is a box describes a step, such as a computation, transmittal of infozmation, or storage or retrieval function. Such flog diagra~ans are read line by line and box by box.
The tenor "party" as generally used herein, indicates an entity, and might be an agent who performs a step or a collection of steps under the protocol. It may also refer to a means or method for perfonszirtg some or all of the steps. Thos, some or all portions of the protocols may be performed wader at~y suitable configuration of digital logic circuitry.
Far example, auy or all steps under the protocol may be realized by not only a general purpose cotaputer, such as a personal computer, but by a hard wired or dedicated combinatorial logic device, or any sort of suitably propratnmed maehiae or naicroprocessar, so long as such device or machine performs the required processing steps, storage, input and output, and the like.
The symbol " Ex " generally indicates that a number or numbers on the le$-hand side sre chosen from a set on the right hand side according to a probability distn'bution that is substantially uniform and independent (random). In practice, a physical random number generator can be used, possibly in conjunction with additior~sl past processing, or a deterministic pseudo-random number generator. Mtthods of generating random numbers are lmown by those skilled in the relevant art.
The symbols "II" and "E" respectively denote product and sum, which are indexed.
The symbol "Zo" denotes a set of numbers of integers 0 through p.-1, or sing of integers, modulo p. Addition and multiplication of elements m the ring Zp are defined modulo p.
The symbol " E " denotes that an element on the left-hand sida is a member or element of a set on the right-hand silo.
The symbol " a " denotes that a set on the left hand side is a subset of a set orl the right-hand side, that is, that the set on the left hand side is contained is the set on the right hand side.

svesT~TUre sHe~ ~RUt_E zs~

WO 01173694 PCT/tTS01/0955D
The angle brackets symbols (t. e., " ( ~ ") are paired symbols that generally indicate that the term or terms between them denote a subgroup generated by a subset of a given group or ring of integers (e.g., the ring Zp). A subgroup is a subset of another group (or ring) that is also a group under the same binary operation (e.g., the integers are a subgroup of the real numbers under addition, but the integers modulo n are not a subgroup of these since the operations are differently defined.
1u the following, unless explicitly stated otherwise, n will be a positive integer, p and g will be prime integers, publicly known.
Arithmetic operations are performed in the r~aodular ring Zp (or occasionally Z"), and g E ZP will have (prime) multiplicative order q. (So, trivially, gl(p-1).) Iri each proof protocol, P will be the prover (shu~ler) and V the verifier (auditor).
One embodilment described below, employs a Zp (F.lGalnal) cryptosystem, although an elliptic curve cryptosystem and cryptosystems under general groups may be used. Such cryptosystems employ public keys for asymmetrical cryptosystems. Public key systems employ so-called one-way functions and trap-door functions. A "one-way function"
is a function that is relatively easy to calculate output therefrom but whose inverse functions are far more difficult to compute. For example, power functions are such that they are easy to compute the product of a number of equal factors, but the reverse operation of finding the root of a quantity, is more complicated. "Trap door" functions are similar to one-way functions, but where the inverse functions are extremely diifficult unless some additional information is available. This additional information is typically the "private key" held by a party, such as a voter.
' The below methods and protocols frequently use the Chaum-Pedersen proof of equality for discrete logarithms. For g, X, h, Y
a Zp this is a proof of knowledge for the relation loggX= loge, Y (1) SUBSTITUTE SHEET (RULE ~6) WO 01/7%9d PCT/US01/09»0 It is not known to be zero-knowledge, however it is known to be zero-laiowledge against an honest verifier, which is su~rient for our main application where the verifier is implemented via the Fiat-Shamir heuristic.
"Zero-lmowledge proofs" allow a voter or proven party to demons>tate lmowledge of a secret while revealing no information whatsoever of use to the vori~fier parry in conveying this demontstration of knoyvledge. Only a single bit of information is conveyed, namely that the pmvcr party actually dots lmow the secret. In other words.-a votez and a verifying authority exchange messages, where the voter's objective is to convince the verifier the truth of an assertion, e.g_, that the encrypted ballot is, or shu$1ed sequence of ballots or elements are, complete and correct, without revealing how die voter voted on each question in the ballot or how the saries of elements were shuffled. Undo such zero..
knowledge proofs, each voter or proven party effectively g~er$tes certain numbers that only a person having his or her own private key could generate. A verifier or authenticating party then confirms that each.
calculated number indeed is generated under a known algorithm to thereby authenticate the voter and that his_ or h~ dectronic ballot is complete and correct or "well-formed" for all allowable choices and constraints, or that the series of elements have not been uttered (besides being shuffled and encrypted).
Definition 1 An instance of the Chaum-Pedersen proof, as above, will be denoted by C'P ($. X. h, ~'!.
Definition Z For feed g ~ ZP be the binary operator on (g~ x (g~ denotes subgroup generated by a s~rbaei of a ring defined by logs ~_ ~a y) _ mgt ~r logsy SUBSTITUTE SHEET (RULE 2s) WO 01/73696 pGT/USOL09550 for all x, y a ~g~ . Alternatively g~ ~ B ~ ~ g~6 - ~)b - ~g~Q
for all a, b a Zq Following the indexing converaionr used for summations and multipltcatiorts, we also trse the notation t ~ ~ X~ = Xo ~= X~ ~s . _ . ~~ Xt ia) This operation as is referred no herein as logarithmic multiplication base Ia each of the notations in the preceding definition, the subscript g may be omitted when its value is clear from context. As generally used herein, "binary operator" refers to an operator defined on a set which takes two elements from the set as inputs and returns a single element.
Remark 1 Notice that in the Chaum-Pedersen proof if h ~, and the eoaunon logarithm is a = logeX= log,,Y, then t:'p (g, X, h, Y) is obviously also a proof of the relation Y = h ~=X ~ X ~l h.
Remark 2 The about proof is obviously zero-lmowledge with respect to s, since the proven need ~nvt have any lalowledge of this value in order to construct the proof.
Note the following collection of well-lmowa results since they will be heavily used in the remainder of the detailed description.
Lemma 1 Let f ~x) a Zq ~:~ , 6e a polynomial of degree d Then " rheie are at most d values z~, . . . , zd a Zq such that f[sr) ~ 0.
Corollary 1 Let~x), g(r) E Z4 [s] be two polynomials of degree at most d, with f ~ g, Thest there are at most d values z~ , . . . , zd E ZQ
such that ,ltzi) = BtZ~).
SUBSTITUTE SHEET (RULE Zs) WO Oa/7369.i PCTlITS01b95.50 Coronary I Lar~~r), g(x) E Z9 L=l ae two p~oly~omials of degree at rnosr d, with f # g. If C E R Zq (c is selected art sad fror~r 2~ ), them the following probability Jalds:
p(~~I(t)=g(~)~ s q Lemma Z Let Z~ be tl~e sk dirnaissional vector space over Z9, mid fcr v = (v1, . . . , . . . v~ E Z~, v $ 0. If r En Zo is chosen as random, then P({r: v.~ =0~~= 1 Q
2. Proofs for Iterated Logari~hmit MultiQlication For the rest of thin section, all logarithmic multiplieations will be computed relative to a fixed element g, and hence we will omit the subscript in notation. The following problem is fimdsmental to the shu$le pmofs which are to come later.
Iterated Logarithmic Multiplication Problem: Two sequences (X, ~; , and ~Y, ) _, are publicly latown. The Prover, P, also Ioaows u, = logy X, and of = logs Y, for all l, but these are unknown to the verifier, Y. P is required to convince V of the relation t t ~X, _ ~Y, (Z) .=y without revealing any information about the secret logarithms r~, and v,.
Intuitively, this pmblem is simple in light of Remark 1_ The Proven can conshuct two sequences of k Chsutn-Pedersen proofs to t - t convince Y of both the value of ~X; and the value of ~Y, , and V can then check that these two values are the same. If all the Xr and Y, are known to be random and independent, this might be acceptable for the purpose of SUBSTITUTE SHEET (RULE Z8) w0 01/73694 PCT/U501/09550 keeping the u, and v; secret and may be implemented under one embodiment of the invention, but it is clear that this protocol reveals some information that v can not compute himself, namely the values of Ir k ~X"~Y" as well as the intermediate logarithatic rnultiplications. In order to strengthen the protocol, the depicted embodiment and method described is detail below introduces some randomness in order to ensure that it leaks no more information than the underlying Chaum-Pedersen protocol.
Iters~ted Logarithmic Multiplicatiott Proof-.
1. P secretly generates, randomly and independently, k+ 1 values {r; }~ a Zq and reveals the exponCntiated values R, = g'' .
2. For each 1 <_ i 5 k, P secretly computes w; = r,.u; /T~, , and reveals W, = g'~ , z; ~ w; /v; .
3. P sets Z; - g'' and constructs the two Chaum-Pedersen proofs CD ~Rr_t.Xr.Rr,W,.~ (3) ~ (Y,g~,R;,Zr~
which he reveals to V. ?here two Chaum-Pedersen proofs taken together cannot reveal any more information about the secrets than the information that is revealed from each of them taken separately. The key to seeing this is Remark 2 The first proof is aero-knowledge with respect to /,r,_, , though only honest verifier zero-knowledge with respect to u; /~;a , But the second proof is zero-knowledge with respect to r, r,_~ and u, since even the Proven need not know these values in order to generate the proof.
Of course, one can gain some information about these values froax the revealed value z" but only if some information is known about v;. Tt is not lmown if this can happen with a dishonest verifier, but does not when SUBSTITUTE SHEET (RULE 26) WO Ol/7369~ PCT/1JS01/09550 the veriSer is honest, and this is the case with embodiments and applications described below.
Clearly the quotients i /r,_, are all uniformly distributed and independent, so the values r, themselves do not reveal any information, by themselves, about the e; and v;.
4, V checks that Z, - g'' and checks each Chaum-Pedersen proof.

5. Y finally evaluates z = ~~~ z; and checks that One can easily check that this protocol solves the iterated logarithmic multiplication problem by referring to Rennark 1 and by simply multiplying out the exponents. The probability of a forged proof is bounded above by the probability that one or more of the Chaum-Pedersen proofs have been forged. Each of these probabilities is 1/q. Hence the ovcrsli probability of a forged proof is bounded above by 2klq.
For reasons that will become apparent later, the following variant of the iterated logarithmic multiplication problem will actually be of more use to the method. _ Scaled Iterated Logarithmic Multiplication Problem: As in the original problem, tyro sequences (X, } :~ and {Y,. }k ~ are publicly known, u, = log~Y; and v, = IogrY~ for all i arc ktwwn to P, but secret from Y In addition, a ~oastant c a 29 is known only to P, but a commitment of c, C = g~ is made known to r! P is required to convince Y df the relation k ~ X; _ ~Y,. (5) r~i ~-i Without revealing any information about the secret logarithms u,, v;, and c.
Scaled Iterated Logarithmic Multiplication Proof:
The proof requires only a miner variation to the original.
The Chaum-Pedersea proof in 4 nccds to be replaced by the similar proof SUBSTITUTE SHEET (RULE I6) G'P (Y;, C,, Wt, Z;) (6) 3. The Simple K-Shuffle The first shuffle pmof we construct requires a restrictive set of conditions. It will be useful for two reason. First it is a basic building block of the more geneial shu$le proof to come later.
Fartuitously, it also serves a second important purpose. A single instance of this proof can be constructed to effectively "commit" a particular permutation. This can be important when shuffles need to be performed on tuples of Zp elements, which is exactly what is required is the voting application. (A "tuple" refers to a sequence or ordered stt. For example, a 5-tuple or quintuple is an ordered set of five alcments, while a k-tuple refers to a finite ordered _set with an unspecifiod number of members) Simple k-Shuffle Problem: Two sequences of k elements of Zr, X,, . . . , Xk and Yl, . . . , Y~ are publicly known. The Proves, P, also lasows u;, = logg.Y; and v, = loggY,, but these are uala~own to the verifier, Y. In addition, a constant c E Z9 is known only to P, but a commitment of c, C = ga is made lrnown to Y. P is required to convince Y that there is some permutation, ~ ~ E, with the property that YA,,} = x,.' ( for all 1 <_ i < k without revealing any information about a or the secret c.
The function re corresponds to a function for mapping a set of input -~ elements to a permuted set of output elements.
Simple k-Shuffle Proof: The proof construction is alutost trivial in light of the previous section and Corollary Z_ 1. V generates s random t E Zq and gives it to P as a challenge.
2. P computes T = g~ (also known T~ and S = T' g".

SUBSTITUTE SHEET (RULE 26) WO OI/7369~1 PC?/U5011095=0 3. P generates the Chaum-Pedersen proof CP (g, C, T, S) and reseals this to V.
4. P computes publicly (i.e., checked by 1~ the values U~ ~ T/X, and Y~ = SIY,. Note: U, and V, are chosen thusly to be more in line with the notation in Corollary 2. In one embodiment, the method implements the protocol with U, m X,IT and Y,- = YdS since divisions are computationally more expensive than multiplicatiorns.
The Proven executes the scaled iterated logarithmic cnultiplieation proof protocol for the sequences {U; ~~~ and { V ~~~ and the comtaitment C_ 8y checking the scaled logarithmic multiplication proofs on sequences U and V, the verifier ensures that the desired relationship between the initial input sequence of elements X and the sequence Y
holds (based on Collorary 2).
A forged proof caa valy be generated if either the scaled iterated logarithmic multiplication proof is fanged, or the single proof of S
= T° is forged, or, V happens to choose c from the exceptional set in Corollary 2. hence the overall probability of a forged proof is bounded above by (3k+ l~q. Further information regardiag proofs provided under the shuffle protocols described hereia ~sy be found in the above-referenced U.S. Provisional Patent Applications_ In ge~aeral, the simple k-shu$le may be sufficient for some applicatioas. To shu$le itetn5, an individual needs to employ a cryptographic transformation (e.g:, exponentiation) where there is certainty that a seriea or sequence of output elements Y~through Yr were derived $rnn as original or iaput sequence of olements Xl through Xk based on wnstant cryptographic information, without revealing which of the original X elements produced a resulting 'Y element_ Furthermore, individuals wish to provide such shuffling without also having to employ a burdensome proof of validity, such as cut and choose type of validity proofs lmown in the prior art that require numerous iterations far a su~cient level of proof. Instead, a series of k independent Chaum-SUBSTITUTE SHEET (RULE 26~

WO 01/7369s ~CTIUSO1/095so Pedersen proofs based on a secret expottentiation value c are employed, as described herein-4. The General R Shuffle An obvious limitation o~ the simple k-ShufFle protocol is that the shu$ler, P, must lmow all the original exponents s,, . . . , s,~. In many applications this will not be the case. The next step is to eliminate this restriction-General k-Shuffle Problem: Two sequences of k elements of Z~, Xl, _ . _ , Xr and Y,, , , , , Yr are publicly latown. In addition, a constant c E ZQ is lotowa only to P, but a wmmittnent of c, C = g' is made lmown to V_ P is required to convince V that there is some permutation, ~c ~ ~ t , with the property that ' Y.ti~ = Xi C$) for all 15 r <_ k without revealing any information about tc or the secret c.
General k Shuf~te Proof. The proof is constructed from a simple k-shu$le that has bees "appropriately randomized" by the verifier, and an application of Lemma 2.
1. P generates a sequence ~u; ~~~ a Z9, randomly and independently and reveals the sequence ~; = g'' .
2. V generates another sequence {e, ~~~ a Zq, randomly and independently, and gives this to P as a challenge.
3. P publicly sets Ui = g' .~I, , and secretly sets u, ~ e, +
.~ u, :, By requiring the Provcr to add a value generated by the Verifier prevents the Drover from picking certaixt secrets (exponents) snd otherovise helps ensure encryption robustness.
4. P cons-nuets a simple k-shn$le on the sequence {U; ~~~ , with another commitment D = ~ (different secret exponent), and inverse permutation, a ~, resulting in the sequence (Y, ~~~ _ ~g"~ }' ~ and the SUBSTITUTE SHEET (RULE 26) Vf0 a117369~t pCTlUS01l09550 corresponding proof as in the simple k-shu$le section. (Recall that the Y
are public, but the v, grc secret to P.) S. P publicly sets A, ~ X w and B, = Y,"' , and reveals the Chaum Pedersen proofs G'P (g. vi, X ~ Ai) CP (g. U~~ Y;. B~ (1~) Thus, tho sequence of elements A and B correspond to the input sequence of elements ?~ at~d Y.

6. Publicly, the values A=~A; (11) t B=~B, (12) are evaluated.

7. P rcv~eals the Chaum-Pedersen proof CP (D, A, C, B) (13}
Steps 5-7 above effectively require the Prover to pin down the data to help ensure that the Prover has not tampered with ~e original data. (These steps differ from the simple shuffle description above.) A
forged proof can only be generated if either the simple k-shufDe proof is forged, or one of the Chaum-Pedersen pmofs is forged, or the tuple (ur,. . .
r~k ) is chosen from the exceptional set in Lemma 2. Hence the overall probability of forgery is bounded above by SUBSTITUTE SHEET (RULE Z6) (3k+1~+2k+1 _(SkT2) (14) 9 q 9 q 5. K-Shuffles of Ta~,les Those skilled in the relevant art will recognize that in the previous section, the choice of the simple shuffle essentially "froze" the permutation that could be proven. This makes it easy to see how to extend the previous section to shuffles of k ruples of elements of ~g~ .
T'hinlcing of a sequence of k I-tuples as a k x 1 array, a single simple k shuffle can serve to prove that all columns have been permuted according to the same permutation, but each row left unchanged. Thus, the k shuffle described above is performed 1 times, once for each column of the array of k elements (each of the k shuffles reuses the simple shuffle). In particular, this extends to tuples of ElGamal pairs.
6. The Voting Application in one embodiment, votes are submitted as ElGamal pairs of the form (g~,ha'm), (or a sequence of these pairs if more data is o9i19i2oo2 09;55 FA.T ~Jo~l WO 01/7369; PCT/USO1lti95i0 requited), where m is some standard encoding of the voter choices (desctt'bed herein), the a; are generated secretly by the voters, and h is a public parameter constretcted via a desIerleSs secret sharing scheme (see, e.g, T. Pedersen_ A threshold ctyptosystem without a trusted party, Advances in Cryptology - EUROCRYPT '9I, Lecture Notes in Computer Science, pp. 522-526, Sponger-Verlag, 1991.), Once the polls are closed (voting finished), an independent collection of authorities sequentially shuffles the ballots. Qn output of the final shu$le, the final collection.of encrypted ballots is decrypted in accordance with the threshold scheme, and the clear text votes are tabulated in full view by normal election rules.
The authorities who participate in the sequential shuffles, may be arbitrary in number, and they may be completely different from those who hold shares of the election private key. The sequence of ballots which are finally decrypted can only be matched with the oarigiaal sequence of submitted ballots if al! of the shuffling authorities collude, since each of their permutations is completely arbitrary_ Each shuffle is performed by an individual authority as follows:
1, ~~ are chosen secretly, randomly and iadepcadently_ 2. Each vote v, _ ( g°' , h~' m ) is replaced, in sequence, by (g'~r~,h~+°m ). A Chaum-Pedersen proof is published without revealing the secrets.
3. A shuffle with secret exponent c is performed on the resulting encrypted votes.
4. Steps 1-2 are repeated.
5. At this point, the messages that are encrypted are the c-th power of ttte original messages. This is easily fixed by raising each coordinate of each vote to the 1!c power. A Chantn-Pedersen proof of this operation is equally easy to provide, thus keeping c secret while convincing verifiers, by simply reversing roles of g and C = g'.
Steps 1 cad 2 above help randomize the input data. to prohibit one from tampering with it such as selecting a relationship SUBSTITUTE SHEET (RULE 26) 09%19/2002 09:55 FAg (~J032 w0 07173694 pCTItJS01109550 between ballots before shu~licvg. The Chaum-Pedersen proof ensures the correctness of the additional value added for this raadamiriag. In an alternative embodiment, steps 1 and 2 may be. omitted (and step 4).
7. Secure Messaee Encoding Ifp-1 has small prime factors, some information about the encoded messages can be leaked. This is not a problem with the shuffle protocols, rather it is a problem for strongly encrypting the m, in the first place. Whether or not this is significant enough to worry about depends on the expected value of the messages to be encoded. 'Ibis problem can also be eliminated however, by taking special care in. the encoding. Each message can be projected onm a subgroup whose order contains only large prime factors. Since most embodiments described herein are restricted to the case where Kg~l is a prime, g, we will only discuss in detail the case p = 2q + 1. However, the more general case can be handled by an extension of these techniques.
Suppose that the bit length of p = 2q + 1 is b, that is 2m <P<2b so 2''~ 5(p-1)~2=q <2°' (15) We require that all messages m be of bit length at most b-2. Rather than enorypting each messagt as ( g° , h°m ) as is standard, we encrypt it as ( 8''. h'mx ) Thus means that all messages are encoded with the same trivial projection on the order 2 subgroup of Z~ . Equation 15 guarantees that the map m -s ma is invertible on the domain of definition. To invert, simply take sUBSTITUTE SHEET (RULE 26) 09'19/2002 09: 5B FAX C~ 033 WO n1/7369a pCT/US01/09550 the unique square root which is less than (p-1)~2. Thus it is a simple matter to recover the original message, m, once the actual message, mz, has been decrypted.

8. Suitable S~rstem Figure 1 and the following discussion provide a brie general description of a suitable computing environment in which aspects of the invention can be implemented. Although not required, embodiments of the invention will be described in the general context of computer-execurable instructions, such as routines executed by a general-pmrpose computer, such as a personal computer or web server. Those skilled in the relevant art will appreciate that aspects of the invention (such as small elections) can be practiced with other computer system configurations, including Internet appliances, hand-held devices, wearable computers, personal digital assistants ("PDAs"), multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, mini computers, cell or mobile phones, set-top boxes, mainframe eocnputers, and the like_ Aspects of the invention can be embodied in a special purpose computer or data processor that is specifically progruruned, configured or constructed to~erForm one or more of the computer-executable instructions explained herein. Indeed, the term "computer," as generally used herein, refers to any of the above devices, as well as arty data processor.
The invention cart also be practiced in distributed computing enviranments where tasks or modules are performed by remote processing devices, which are linked through a communications network, such as a Local Area Network (LAN), Wide Area Network (WAN), or the Internet. In a distributed computing environment, program modules or sub-routines may be located in both local and remote memory storage devices. The invention described herein may be stored or distributed on computer-readable media, including magnetic and optically readable and removable computer disks, stored as fumware in chips, as well as SUBSTITUTE SHEET (RULE 26) 09/19i2002 09:58 FAg f~034 WO OI!'7369d PCTJUSOI/09530 distributed electronically over the Internet or other networks (inducting wireless networks). Those skilled in the relevant art will recognize that portions of the protocols deseiibed herein may reside on a serves Computer, while corresponding portions reside on client computers_ Data structures and transmission of data particular to such protocols are also encompassed within the scope of the invention.
Unless described otherwise, the construction and operation of the various blocks shown in Figure 1 are of conventional desigi. As a result, such blocks need not be described in fiuther detail herein, as they will be readily undesstood by those skilled in the relcvsnt art.
Referring to Figure 1, a suitable environment of system 100 includes one or more voter or client computers 102, each of which includes a browser program module 104 that permits the computer to access and exchange data with the Interact, including web sites within the World Wide Web portion 106 of the Internet. The voter computers 102 may include one or more central processing units or other logic processing circuitry, memory, input devices (e.g., keyboards, microphones, touch screens, and pointing devices), output devices (e.g., display devices, audio speakers and printers), and storage devices (e.g., fixed, floppy, and optical disk drives), all well known but not shown in Figure 1. The voter computers 102 may also include other program modules, such as an operating system, one or more application programs (e.g., word processing or spread sheet applications), and the like. As shown in Figure 1, there are N number of voter computers 102, x~epresentiag voters 1, 2, 3 . . . N.
A server computer system 108 or "vote collection center,"
coupled to the Internet or Wvrld Wide Web ("Web") 106, performs much or all of the ballot collection, storing and other processes, A database 110, coupled to the server computer 108, stores much of the Web pages and data (including ballots and shu$le validity. proofs) exchanged between the voter computers 102, one or more voting poll computers 112 and the server computer 108. The voting poll computer l I2 is a personal SUBSTITUTE SHEET (RULE 2s) computer, server computer, mini-computer, or the like, positioned at a public voting location to permit members of the public, or voters who may not have ready access to computers coupled to the Internet 106, to electronically vote under the system described herein. Thus, the voter computers 102 may be positioned at individual voter's homes, where one or more voting poll computers 112 are located publicly or otherwise accessible to voters in a public election. The voting poll computer 112 may include a local area network (LAN) having one server computer and several client computers or voter terminals coupled thereto via the LAN to thereby permit several voters to vote simultaneously or in parallel.
The voting poll computer may also include an rButton reader for reading iButtons, such as cryptographic iButtons provided by DaTIas Semiconductor Corp. An rButton is a 16 mm computer chip armored in a stainless steel can that may include a microprocessor for high-speed arithmetic computations necessary to encrypt and decrypt information, such as signatures of voters who have registered.
_ Of course, other data storage devices besides iButtons may be employed, such as computer readable media described herein, radio frequency identification (RFID) tags, one or two dimensional bar codes or other data collection devices, and associated readers for these.
Under an alternative embodiment, the system 100 may be used in the context of a private election, such as the election of corporate officers or board members. Under this embodiment, the voter computers 102 may be laptops or desktop computers of shareholders, and the voting poll computer 112 can be one or more computers positioned within the company (e.g., in the lobby) performing the election. Thus, shareholders may visit the company to access the voting poll computer 112 to cast their votes.
One or more authority or organization computers 114 are also coupled to the server computer system 108 via the Internet 106. The authority computers 114 each hold a key necessary to decrypt the WO 01173694 PCr/USOLO956o electronic ballots stored is the database 110. Threshold crypt<rgraphic systems require that a subset t of the total number of authorities n (i.e., tin) agree to dectypt the ballots, to thereby avoid the requirement that all authorities are seeded for ballot decryption. 1n other vYVrds, the objective of s threshold cryptosystem is to share a private key, s, among n members of a group such that messages can be decrypted when a substantial subset, T, cooperate - a (t, n) threshold aypbosystxm. Protocols are de$ned to (1) generate keys jointly among the group, and (2) decrypt messages without reconstructing the private key. The authority computers 11A may provide their decryption share to the server computer system 108 after the voting period ends so that the server eoraputer system may decrypt die ballots and tally the results.
Furthermore, undo the depicted embodiment, each of the authority computers perform one shu$le of the ballots, as described herein. In conjunction with each shuffle, each authority computer generates a shuffle validity proof, which may be encrypted cad forwarded to the server computer 108, or stored locally by the authority computer.
In as alternative embodiment, an additioasl set of authority computers are provided, where one act of authority computers shu$le the encrypted ballots and generate shuffle validity proofs, while the second set of authority computers employ keys shares for decrypting the ballots.
One or more optional verifier computers 130 may also be provided, similar to the authority computers 114. 'The verifier computers may restive election transcripts to verify that the election bas not been compromised. For example, the verifier computers may receive the .shu$le validity proofs from each o~ the authority computers, as described herein. The verifier computers may perform verifications aft~a the election, and need not be connected to the Internet. Indeed, the verifications may be performed by other coraputrss shown or described herein..
the server, verifier or authority connputers rnay perform voter registration protocols, or separate registration computers may be SUBSTITUTE SHEET (RULE Z6) w0 01/7369; PCT/ITSO1/0955D
provided (not shown). The registration computers may include biometric readers for reading biometric data of registrants, such as fingelpaitlt data, voice fingerprint data, digital picture comparison, and other techniques lozown by those skilled in the relevant art. Voter registration and issuing anonymous certificates for use with ve 'nflable shuffles is described below.
The seer computer 108 includes a server engine 120, a web page management component 122, a database management component 124, as well as ottrer components not shown_ 'fheserver engine 120 performs, in addition to standard functionality, portions of the electronic voting protowl. The encryption protocol may be stored on the server computer, and portions of such protocol also stored on the dient computers, together with appropriate constants. Indeed, the above protocol may be stored and distributed on coxaputer readable media, including magnetic and optically readable and removable computer disks, microcode stored on semiconductor chips (e.g., EEPRQIVn, as well as distributed electronically over the Internet or other networks. Those skilled in the relevant art will recognize that portions of the protocol reside on the server computer, while corresponding portions reside on the client computer. Data structures and transmission of data particular to the above protocol are also encompassed within the present invention. Thus, the server engine 120 may perform all necessary ballot transmission to authorized voters, ballot collection, verifying ballots (e.g_, checking digital signatures and passing verification of included proofs of validity in ballots), vote aggregation, ballot decryption and/or vote tabulation. Under an alternative embodiment, the server engine 120 simply collects all electronic ballots as a data collection center. The electronic ballots are then stored and provided to a third party organization conducting the election, such as a municipality, together with tools to shuffle ballots, decrypt the tally and produce election results. Likewise, election audit inforiaation, such as shuffle validity proofs and the like may be stored locally or provided to a municipality yr other organization.
SUBSTITUTE SHEET (RULE 26) WO 01/73694 PCT/US01109S5o The web page component 122 handles creation and display or routing of web pages such as an electronic ballot box web page, as described below. Voters sad users may access the server computer 108 by mesas of a URL associated therewith, such as htxp:\\w~uvw.votehere.net, or a URL associated with the election, such as a URL for a municipality. The municipality may host or operate the servez computer system 108 directly, or automatically forward such received electronic ballots to a third patty vote authorizer who may operate the server computer system. The URL, or any link or address noted herein, can be any resource locator.
The web page management process 122 sad server computes 108 may have secure sections or pages that may only be accessed by authorized people, such as authorized voters or system administrators. The scryer computrr 108 may employ a secure socket layer ("SSL") and tokens or cookies to authenticate such users. Indeed, for small elections, or those where the probability of fraud is low (or results of fraud are relatively inconsequential), the system 100 may employ such simple network security measures for gathering and storing votes as explained below, rather than employing complex electronic encrypted ballots, as described in the above-noted patent application.
Methods of authenticating users (such as through the use of passwords), establishing secure transmission connections, and providiryg secure servers and web pages are lmown to those skilled in the relevant art.
The election scheme and system uses a "bulletin board"
where each posting is digitally signed anal nothing can be erased. See papers by K Sako, J. K~iia~ R. sad Cramer, R. Gcnnaro, B.
Schoenmakers_ The bulletin hoard is implemented as a web server, The "ballot box" resides on the bulletin board and holds all of the encrypted ballots. Erasing can be prevented by whiting the web server data to a write-once, read-many (WORM) permanent storage medium or similar device. Further details on such a bulletin board system are found in U.S.

SUBSTITUTE SHEET (RULE 26) WO Ol/7369~ PCT/IlSo1l095io Patent Application No. 09/534,836, entitled "Electronic Voting Scheme Fsaploying Panaaneat Ballot Storage."
Note that while one embodiment of the invention is described haeia as employing the Iatemet to connect computers, other alternative embodiments arc possible, For exsraple, aspects of the invention may be employed by stand alone computers_ Aspects of the invention may also be employed by say interconnected ilam processing machines. Rather than employing a bmwser, such machines may employ client so$ware for implementing aspects of the methods or protocols described heroin.

9-EI ct~on Expmple One application of the gaieral k shuttle protocol is in the area of electronic voting. In order tA make an election universally verifiable, submitted ballots must initially be irrefutably connectable to a valid (i.e., registered) voter. Somehow ballots must be "separated from their signatures" by a verifiable process - i.e., one that does not allow phony ballots to be substituted in the separation process - before they can be "opened".
The protocol we present here relies on a set of N "tabulation authorities" with differing interests is the election results.
1. The protocol is a threshold scheme in that at least t aufhorities must behave honestly in order for tabulation to be completed.
(The parameter t can be chosen in advance of ~e election~to be nay value 1 S r S N.) Thus, in paxticular, it is not necessary that the authorities complete their shuffles in any particular order, nor is it even necessary (except in the special case t ~ N) that all the authorities participate.
2. Even if all the authorities conspire, they cannot produce false election results without being caught by as external auditor who wishes to verify the rtsults 3. Privacy of sa individual vote catv only be compromised by a conspiracy of at least r of the authorities to do sa_ SUBSTITUTE SHEET (RULE 26) wo oin3s9a pcr~rsoiro9sso The protocol proceeds as follows:
First: Initialize Election 1. The sutborities first agree on the election parameters:
(a) Parameters necessary for any election, including: the set of eligible voters, the ballot duestions, the ballot answers, the ballot style, the time the polls are to be opened and closed, ctc.
(b) The collection of tabulation authorities: i.e., themselves. (We henceforth use N to denote the number of authorities in this group.) (c) .~ The threshold parameter, 1.
(d) A shuffle parameter, 1 ~ a S t. (s = t is a natural choice.) (e) A group G and a subgroup generator, g a G.
(In order to achieve secure encryption, the prime factors of ~ should be large, however, this requiranent is, of course, open to interpretation by the autharities thennselves.) (fj A standard "bit encoding" for responses) (e.g_, ASCII) and a small "message multiplicity" integer d Z 1. The message multiplicity refers an agreed upon subdivision of each ballot, and corresponds to ballot formatting (similar to the layout of a paper ballot to indicate where responses to each ballot question are to be located). d is typically chosen as small as possible to SUBSTITUTE SHEET (RULE 26) accommodate the ballot response size. Most often, d = 1 will world because the message multiplicity is determined by the key lengdz, and because a sufficiently large key length (e.g., 1024 bits) can accommodate most baDots having a ressonable number of questions.
(g) A group element h a (g) which is created by way of an (N, t) - secret clearing scheme executed by the authorities. (See, T. Pedersen article.
2. Once agreennent is reached on the election parameters, the authorities all "sign" them, and some representation of this signed set of parameters becomes the signed ballot, Second: Vote 1. During the election (i.e., while "polls are open"), each voter V . encodes his responses) by the election standard "bit encoding" (agreed upon and signed by tie authorities during election initialization - see above) into a sequence of messages, m~,.." nod a G.
(More on this in section below.) The "message multiplicity," d is another election parameter (see above).
2. Y selects exponents cr,,...,ad independently at random from 0 S a~ < ~ for the encryption.
3. Y returns to the ballot collection center, the encrypted response sequence (8''> h°'m; )...., (8°'> ha', ma) (16) along with an "attached" digital signature, created by Y, to authenticate the response by tying it to a particular eligible voter.

SUBSTITUTE SHEET (RULE ~6) wo oir~~ rCrmso~ssu 4. If the digital signature submitted by Y verifies and Y
has not previously been issued a ballot, then Y is issued a receipt, sigied by the cenn~al collection agency. This receipt ac3anowledges that a ballot from this votes, in this particular election has been received, but includes no information (not even encrypted or hashed information) about the contents of the ballot. 'the receipt may also be broadcast to the voter.
The receipt also serves to confirm that the voter's ballot was not lost and not maliciously deleted.
Thlrd: Tabulate Results 1. At the start, the collection of voter responses are laid out in 2d sequences, each of length N~,, where N~,sr is the total number of ballot responses rcccived. Fach sequence corresponds to a coordinate in the standard ballot response format (equ~ttio~n ~ø). The entries is each sequence are ordtrcd by voter. The index assi~ed to each voter is not important, just so long as the indicts are consistent, This way, an external observer can check that the signed ballots have been transformed in a very simple way, and that, applying the night interpretation to the data layout, it still represents the same set of responses that were signed arid received.
2. In any convenient order, a sequence of s authorities each execute the following steps:
(a) Let S be the authority currently in sequence.
(b) S selects independently at random aBV~"
exponents I<_~gf~c~g) lSj_<N~"andl5l~d (c) S calculates ~e group elements gs v~ n = g°p sad hr (j, ~ = i~~~' . further, as intermediate Chaum-Pedersen proof is generated on (g, g~( J~ ~~ H~ hs~J~ ~)..

8UB5TITUTE-SNEET (RULE 26) WO O1/7369.i PCT/US01/09550 (d) S then transforms the 2d input sequences into 2d intermediate sequences. The j-th entry of the 1-th input sequence of the form ga m is transformed by multiplying it by gs ~', ~ and the j-th entry of the 1 th input sequence of the form h°ni is transformed by multiplying it by ha (j, Q. The transformed entries are ail. kept in the same order.
(e) S chooses a random exponent 0 5 c < fig/, and a random permutarion fi a ~N~, and commits ha = g'.
(f) S then executes a geaeral k-shuffle (with k =
N~) on each of the 2d intermediate sequences, using the secret parameters c and 9r1, and rearing the same simple k shuffle as the basis for each general k-shuffle. (This ensures that each of the Zd sequences are subjected to the.same secret "permutation".) (p~ (i) S repeats step (d) with new random pi's.
(ii) raising each coordinate of each vote to the 1/c power and providing a Chaum-Pedersea proof of this operation, thus keeping c secret while convincing verifiers, by simply reversing roles of g aad C = gs:
(h) (Note that S need not explicitly compute the intermediate sequences at this stage. ?hey are necessary for external verification later, but the output can be computed directly and the intermediate sequences constructed on request SUBSTITUTE SHEET (RULE 26) of an auditor. However, security concerns may dictate that the auditor perform the verifications before beginnnng the next shuffle.) 3. Shu$led ballots are now reconstructed by combining entries of each of the 2d sequences ia~ the same way they were formed.
4. Finally, t authorities execute the threshold decryption protocol on each shu$led ballot.
In general, the tabulation phase includes two subphases.
First, a sot of T _< t of the tabulation authorities each execute, in sequence, a verifiable k x d shuffle (where k is the total number of ballots cast).
The output sequence and proofs from each shuffle is signed and passed to the next tabulation authority. (Each tabulation authority executes its shuffle only if the input Sasses both a signature check and a check of the (previous) shuffle zero-latowledge proof ("ZKP") and the intermediate Chautn-Pedersen proofs.) Second, once the full round of shu$les have been executed and verified, a set of r tabulation authorities use their secret key shares to jointly (and verifiably) compute a decryption of each m's in the final set of filCiatnal pairs.
In general, each shu$ling authority will latow the input-output correspondence, since it is responsible for generating the permutation in the first place. However, shuffles are staged. Thus the output of one shuffle is used as the input to another shu$le performed by a different shuffling authority. Thus, unless all authorities conspire, no one shu$ling authority will have any lmowledge of the correspondence . between initial input and Seal output, As described below, however, a further enhancealeat eliminates this possibility.
Faurth: Externally Verify Elecf9on On request, each authority publishes (a) His interrnediatc sequences.

SUBSTITUTE SHEET (RULE 26) (b) Chaum-Pedersen proofs P(g, g,( j, Tj, h, h,( f, 1)) for 1 5 j S N~" and 1 S 1 _< d.
(c) His k shuffle proof, (d) The Chaum-Pedersen proofs under step (g) about.
In general, an election trrorascrtpr may be published that contains the following:
1. The voter roll contaiaiag voter identification information and voter public keys_ 2. The otigiaal set of k voter-signed ballots_ 3. The t k x d shuffles (including proofs, as noted above.
4, The final share decryption validity proofs.
5. The final tallies.
Remark: Several variations oa the order in which the authorities execute their tabulation steps (Tabulation steps 2 (a) - (h) above) are possible. Is particular, the steps can be interleaved under alteraati~re embodiments.
Remark: The External Verification phase can be carried out as tabulation is going on, or at a later time. The authorities need only save their stage parameters.
Refcrrsng to Figgie 2, a schematic diagram illustrates a basic application o~ the shu$le protocol to an election, shows as a method 200. In block 202, three eaaypted ballots are snbmiited, one each for voters Joe Smith, Sally Jones, and Ian Kelleigh. In block 204> the list or roll of voters is separated from the encrypted ballots, which are shown in block 206. Thereafter, a one-way reencryption of the ballots is performed m produce a shuffled set of ballots, shown in block 208. A shuffle SUBSTITUTE SHEET (RULE Z6) w0 01/73694 PC?lfSO1l09550 validity proof is generated based on this first shu$le, shown in block 210.
?he shuffle validity proof allows a third party to ensure that all input data (the ballots) had the same operation applied to them, and that no altering of the ballots had been performed.
A second shuffle of the (previously shu$led) ballOtS 15 performed, to generate a second shu$led set of ballots, shown as block 212. Again, a shuffle validity proof is generated, shown in block 214.
The shuffled ballots of block 212 are shuffled a third time, to produce a final shu$led set of ballots under block 216_ A third validity proof 218 is likewise generated based on the third shuffle. In. sum, a three-by-three shu$le array is provided under this example. Following, the shuffling, the ballots are decrypted to product a tally, shown as block 220. A third parry rnay verify that the election by analyzing, among other things, each shu$le validity pmof to ensure that each shuftler has preserved election integrity.
The shuffle protocol is presented above as effectively separate subroutines that may be employed for various applications, such as in a electronic voting scheme. A first subroutine provides the fiu~ctionality of scaled, iterated, logarithmic multiplication pmofs between a proven and a verifier. A second subroutine provides the functionality of a simple shu$le protocol and employs the scaled, iterated, logarithmic multiplication proofs. Thereat~ter, a third subroutine implements general shu$le fuactio:<ality, where the shuffler does not lmow the exponents, building upon the second subroutine of the simple shuffle. A fourth sub~routinc extends the third subroutine to shu~Iing k tuples of elements.
~ w Referring to Figure 3, a routine 300 is shown for implementing scaled" iterated, logarithmic multiplication proofs. In block 302, initial cryptographic parameters are agreed upon, such as by a voting organization_ These initial parameters include the group (e_g., ZD), a subgroup operator g, the size of the group G, and the size of the generated subgroups p and q. This information may be provided to a number n of shu$ler or authority computers 114 and verifier computers 130.

SUBSTITUTE SHEET (RULE Z6) WO 01/7369 PCT/U'S01/09550 In block 304, the shuftler (or Drover P) selects a secret exponent c, and based on the subgroup generator g generates C.
Additionally, the shu~ler may receive or generate Y values for received X's and, for indexes of i for 1 through k, and provides the Xs, Y's, and C
to the verifier_ In block 304, the shu8ler also seerady generates random exponents as ri, which, based on the subgroup generator g, are used to generate valves R~ for each value of i of 0 through k. Similarly, under block 304, the shu$ler employs the generated random exponent r, to gtnerate Wi and Z,.
In block 306, the shu~ler provides Chaum-Pedersen proofs for each. clement 1 through k for the values of Rte,, Xw R;, W,, and Yw C, Wb Z,. These values for the Chaum-Pedersen proofs are then provided to the verifier, together arith values z, and Ro. The vezifier then, in block 308, verifies the correctness of the proof data to accept or reject the proof Ia other words, the verifier checks that each z, as as exponent to the subgroup generator g, generates a corresponding Z, checks each Chauro-Pedersen pmo~ checks that the product of the zf s is equal to z, and that the value RD raised to the power z is equal tctRk.
Referring to Figure 4, a routine 400 is shown for performing a simple shuffle protocol, as described above. Following block 302, the block 404 is similar to block 304, but the shu$ler shuffles the X eleaaents by a permutation a to generate the Y elements. 'Ihe verifier in block 406 generates a random value t as a ehallenge_ In response, the shuffler is block 408 uses t as an oxponem to the subgroup generator g to secretly generate the value T, which, when combined with the shuffler's secret exponent c, permits the shu$1er to secretly generates a value S. As shown, the shuftler they publicly generates values U and v and provides a Chaum-Pedersen proof for (g, C, T. ~ ~ block 410. In block 410, the shu$ler also generates proof data as scaled iterated logarithmic multiQlication proof for each of the elements X and Y in the series of i of 1 through k. The proof data is then provided to the verifier in block 412.

SUBSTITUTE SHEET (RULE Z6~

we oin3s9s pcrmsovo9sso The verifier verifies the correctness of the proof data and accepts or rejects it In other words, the vaiBer executes the scaled iterated logsrithtnic multiplication proof protocol noted above for ~e sequcrnces of U and V, and checks the commitment value C_ Referning to Figure 5, a general shu$le protocol SOD is shown where the shu#ler does not know the exponents. The initial steps in the protocol 500 are similar to that of 400, except that the verifier adds a raado~,~ element to the shu8ler's secret exponents. As shown in block 502, the shuffler secretly generates a random sequence of initial values, which are used as exponents with the subgroup generator g to generate an initial sequence (t7! =g' ). Likewise, in block 504, the veri5a secretly generates another seqneaee of elements a for values l of 1 through k, and provides the sequence to the shu$ler as a challenge. In block 506, the shut~ler secretly adds the sequence challenge a to the previous sequence, to then publicly generates a series of values t! (U, _ g4U! ~.
In block 508, the shu$ler constructs a simple k shuffle on the sequence U with another secretly generated ~ conzmitmcnt D (that employs a different secret exponent d chosen by the shu$ler) and generates a sequence of values V Then publicly, the shu~ler reseals t,'.hautn-Pedersen proofs for a sequence of values A and B for indexes 1 through k, publicly generates the product of the sequences as values A
and B, and provides a Chaum-Pedersen proof for the relatiaa between D, A, C and B, as shown. Under block 510, this proof data is provided to the verifier, who verifies it under block 512.
_r ~~

10. Issuino norvmous Certificates With Verifiable Shuf~lea Presented above is a new, efficient construction for veritably shu$ling encrypted data, and a particular way that this construction can be used to conduct a ~rnivarsally verifiable electronic election system. That system depends on a collection of election authorities to "shu$le," or "anonymize" the ballot data that has been SUBSTITUTE SHEEP (RULE 16) WO 01/73fi9~ PCT/USULU95S0 collected at vote time. ?his process takes place after aU votes have been cast, but before ballots am decrypted and tabulated, The validity construction prevents any one or more of the election authorities from making any changes to the original election data without being discovered by anyone auditing the final election transcript.
A disadvantage with this approach is that voter anvnymiry is not protected by as strong a mechanism as is election integrity. Election integrity is protected by pure computational intractibility-it is essentially impossible for the election authorities to produce false election results without detection-even if they arll act in collusion. However, by acting in collusion, they are able to determine the contents of any individual voter's ballot with relative case.
The same underlying shuffle construction can be used to construct a new election protocol that eliminates this weakness. The idea is to awve the shuffling to the registration, or ballot request phase of the election, thereby anonynuzing the identities of the voters without losing strict control, and audit of election eligibility rules-i_e., only ballots cast by registered voters should be counted, and multiple ballots from the same voter should not be accepted. With. this accomplished, it is no longer avert necessary to encrypt ballots, and tabulation can be performed "in the clear =which is obviously an easy process to audit.
The idea of using the construction as part of an anenyrnous registration process has applications beyond voting. Any situation where access to a resource, such as a server or file, needs to be limited to authorized personnel, but where those who are authorized wish to protect their individual identity, may use this wnstruction to meet both requirements simultaneously. For example, applications of group signatures may be equally applicable to the protocols described herein.
Note also that the term "voter" is generally used herein to refer to any individual or organiizarion that employs some or all of the protocols described herein.

SUBSTITUTE SHEET (RULE Z6) WO Ol/7369~ PC1'1U501/095.50 Outline of the Protocols Two protocol variants are provided, both of which follow the same high level flow of information. Each protocol begins with the assumption that a set of public keys has been stored in some central authentication database, or certificate server. Each corresponding private key is known by one, end only one, eligible votes. Furthermore, the correspondence between public key sad individual voter is known by the entity, or entities, who control the certificate server. ('The exact form of these public/private key pates are slightly different in each variant of the protocol.) In practice, the public keys will likely be wrapped in the form of a digital certtfircate which ties ate idenbfjning information together with the public key in a single piece of formatted data (This is the convention followed by widely accepted Public Key Infrastructures, or PKTs.) Typically, this distribution of keys and certificates wiD be accomplished by a tightly controlled registration process, the most secure of which would be an "in person" registration process where the voters can be physically identified at the time of certificate generation. (Suvh registration processes are described in detail in U.S. Patent Application No. 09/534,836 noted above.) It is important to distinguish between two di$erent types of ccrbhcates that exist in the protocols. The first type are the certificates just described, where the association between public key and individual person is publicly, or at least widely known ("standard certificates"). The second type are the certificates that will be distributed in the stages of the protocol that follow the initial registration phase just described ("anonymous certij~cates"). These anonymous certificates are -distinguishable from each other, at very least by the fact that they contain different public keys, however, the only individual who knows the owner of a gven anonymous certificate is the owner himself. It is the goal of the protocol to guarantee that ~ Only individuals who own one of the standard certificates are issued an anonymous certtf' tcate.

SUBStITUTE SI~IEET (tZULE 261 WO Ol/'f369-~ PC't/I1S01/U95.50 In most applications, such as ~e voting application, it is also the goal o~ the protocol to guarantee that Each individual is issued only as many anonymous certificates as he/she has standard certiscates. (Usually, each owner of a standard cerdScate v~n'll have only one standard certificate) Once the registration of standard ceriiScates is cotnplLte, the protocol variants each proceed as follows.
Initisliretion: A set, K, of raw public keys is corutructed at the certificate server (e.g., server 108), and initially set to be the set of public keys associated with the set of standard certificates. The set of public keys is generated during the initial registration process of each individual, when that individual registers and receives, for example, a standard digital certificate. ?he public keys generated under the initial registration process are pooled together to generate the set X. Each individual holds a private key associated with one of the public keys is the set K.
1. An individual, P, contacts the certificate serv~ei; S, through a digital communication channel (such as the Internet) indicating that he wishes to obtain au anonymous certificate.
2. S ret~nns to P a sat, H a K , of public keys (which includes Ss public key), and stores the set J = 1i' - H . (Ideally, H = X and J = 0 , but for reasons of communication bandwidth, the inclusion may be proper. For example, a subset of the public keys K may be provided to the individual P where the set of public keys is quite large, and bandwidth constraints for transmission effectively Iimit transmission of such a large set of keys. For other reasons, SUBSTITUTE SHEET (RULE is) WO oill3s9; pCTlUSo11o9550 the certificate server may wish to return only a subset of the public keys.) 3. P selects a subset M ~ H , which may be all of H, and sets M'~H-M
4. P computes X' which is a shuffle transformation ofM_ (See above and the following sections.) P also generates a formatted anonyntovs cent fcate requesr_ 'This is done by generating a random public/private key pair, and formatting tho public part with some "random )D" data to conform to a specified certificate format. (Needless to say, P must also store the private part in some safe place.) 5. P returns X ', M and M' to S along with (a) The shu, j'le transcript, or validity proof, wrhieh proves to S, or any ar~ditor, that X' is, in fact, a valid shuffle transformation ofM.
(b) A proof that P knows the private key corresponding to a particarlar element, h a fl' (c) The formatted certificate request.
6. S checks that H = M v M' along with the validity of both Sa and Sb.
(a) If ar~yr of the checks fail, S indicates failure to P and either terminates the contircunication with P, or gives P an appropriate chance to retry_ (b) If both checks pass, then SUBSTITUTE SHEET (RULE Z6) WO Oi17369.t pC"I/US01I095E0 i. . If anonymous certificates are intended to be issued only a»ce to each owner of a standard cert~cate, S sets g~~' H~_~~ (17) x=.r~~.r~~xw (1g) or, if, for some reason, it is desired to issrce anortyno~rs cert~cates nrulttple times to each owner of a standard carti~cate, S sets x =r~~.r~~x~ (19) ii. Anc~ S digitally signs the formatted certif'~rcate request thereby creating an anareyrnorcs certificate-returtrs the (signed) certificate to P, and stores tits certt,Jlcate in the data base for later anonymous authentication.
7. The pmcess now continues from the beginning with a new P, and x appropriately modi5ed.
In other words, the individual P may prove to the certific$te server S that the individual holds a private key associated with one of the public keys is the subset M selected by the individual, without revealing which one by shu$ling the subset M of public keys. After issuing an ano~rmous certificate, the certification setvrer then removes the one shuffled public key from the shu#lled set of public keys for use by the next iadividusl requesting an anonymous certificate.
Anonymous Authentication and Sigoatur~ea The basic construction a~f the shu$le protocol above solves the following problem.

SUBSTITUTE SHEET (RULE Z6) wo oirrssss pCTIUSOi/o9sso General It Shuffle Problem: Two sequences of k elements of ZP, S = ~X,,...,~L'k ~, and T = ~1;,...,Yt ~ are publicly laiown. Is~
addition, ~t constant c a Zq is lmown only to P, but a commitment of c, C = g° is made known to V P is required to convince V that there is some permutation, st E ~k , with the property that Y"~r~ = X; (2Q) iwr all 15 t 5 k without reveali:lg stay information about ~c or the secret c.
In the shuffle protocol above the solution to this problem is first presented as as interactive proof protocol executed by P and V, but it is made non-imeraetive by standard application of the Fiat-Shamir heuristic. We denote the resulting verification transcript, produced by the shuffler, P, by T (S, T, g, .C7, Anonymous A~rthentication Protocol 1 rn this variiant of the protocol ~ the public keys are elements h E (g) c Z, , and the eorr~sponding private keys are simply the socaret exponents, s = lost h , ~ The set H must always be taken to be all of K, i. e. H = K .
The set M must also always be all of H, i. e. M = X and M'=o.
. S must store one additional modular integer, G a ~g~, which will be modified duriag each autlneatication session. At iaitislizatioa, G is set equal to g_ The protocol proceeds as described in the previous section, with the following modifications_ 1. In step 2, S must also return G to P.

SUBSTITUTE SHEET (RULE 26j WO 01/'7369; PCZYtJ501/09550 2. The tcsnscript that is retuincd to P in step Sa is exactly T(M,H',G,C)=T(H,H',G,C) (21) 3. The proof of private key latowledge in step Sb, is exactly the integer a = cs ~ Z9, along with the particular value h' a H' (or its index) which satisfies h' = G' (22) Note that there will be one, and only one, such value.
Further note that since c is random and independent of s, revealing a does not reveal information about s. The corresponding check that S perfvtms is simply to verify equation 22.
4. 1~f the checks in equation 22 all pass, then in additiozt to the transformations performed is 1 and 2, S also performs the transformation G = C (23) Anonymous Autbentication Protocol Z
A shortcomiltg of the first anonymous authentication protocol is that the set to be shuffled by P must always be all of X. The same transformation (exponentiation) is applied to all public keys in the set H~K Since tech of the transcripts T(H,H',G,C), must be stored until all audit requirements are fulfilled, this can create a large amount of data if the original set of standard certificates is large. This problem is addressed with the following second anonymous authentication protocol.
In this variant of the protocol SUBSTITUTE SHEET (RULE 26) w0 oinss9s PCTIUSoi/o9550 The public keys are pairs of elements (k, h) E ~g~ x (g), and the corresponding private keys are simply the secret exponents, s = log k h .
Ihc set H must contain P's public key. This can be achieved in s variety of ways.
1. S and P can engage in a series of retries until this property ofH is achieved Z. or, at initial registration, standard certificates can be assigned to "blocks. " When P first contacts S, he identifies himself only so far as his block number.
Effectively, a difFereat base G is act for each individual P, aad the individual shuffles only a subset of the entire set of public keys (which subset includes the voter's public private key pair). The protocol proceeds as described in the previous section, with the following modifications.
1. The transcript that is returned to P in step Sa is the shuffle transcript fvr the set of pairs. See above for the details of this construction.
Z. The proof of private key lmowledge in step Sb, needs to be a bit more complicated in order to avoid revealing the private key.
(a) P must Indicate to S a particular pair, (k', h'~ a H', or its index, which is the new index ojthe pair belonging to P's private key. That is, h' _ (k ~l.
(Notice that such a pair exists uniquely since the shu,8?e operation esponentiates both the k's and the h's Iv the same secset ezponent c. So h=k' if, and only if, h' _ (k')' .) SUBSTITUTE SHEEP (RULE 26~

WO 01173691 ptTJt1S01l09550 (b) P reveals to S a 'hero-~owledge proof that he, P, knoyvs s = logs h : (See the Chaum articles,) This proves that P knows the corresponding private key without revealing it.
3. The corresponding checks that S must perform are obvious.
(a) S checks the varlidity of P's shrr~le trr~nsc~pt . , (b) S checla the validiry of P's proof that he known that s = lo8a~s.
Note: under an alternative embodiment, some or all of the keys in the set K (1.e., the subset ~ may be shuffled by certain individuals or authorities before any one individual requests an anonymous certificate. In other words the pool of public keys stay be sufficiently ra:tdomized before eithtr of the above anonymous authentication protocols are employed for a particular requesting individual. As s result, a smaller subset of public keys may be selected by each individual under Protocol 2_ Referring m Figure 6, an ex~nple of a routine 600 for implemtenting the first variant of the anonymous certificate distribution is shown. After initializing cryptographic parameters is block 302, a stutdard set of public keys H are provided, in block 604, which may be collected by a registration set~rer after a set of registrants or voters have each registered and submitted public keys h (that comspond to individually held private keys s, as shown in block 606). In block 60B, the subgroup geue~ g is set to G.
In block 610, an optional randomization performed by one or more authorities may be performed. Under block 610, in sequence, each authority perfornas a verifiable shuffle on the set of standard public keys H nsiag (G, C-G'~ as a shu$le commitment, where c is a secret key held by the authority. Each authority rchans the shuffled set of public keys, H', along with shuffle verification transcript, T (H, H', G, C~ by SUB8T1TUTE SHEET (RULE 26) WO 01/7369a PCT1U5o1/o9550 employing the general shuffle described above. If the verification transcript is correc'f, then the registration server performs the substitutions G~C and HsH; and stores the previous values, along vrith the shu$le verification transcript for later auditing purposes_ The optional randomization under block 610 may be performed as part of the previous initialization, or at any intermediate stage of anonymous certificate generation described below.
Blocks 612-618 represent a single request for an ananymous certificate by an individual who previously provided one of the public keys h in the set H. These steps are repeated for each requesting registrant. In block 612, the registrant (or more accurately, the voter's computer 102) generates a request for an anonymous certificate. In response thereto, the registration server retrieves the value G , and the set of public keys H under blocks 6I4 and returns them to the registrant. 1n block 616, the registrant computes a shuffle and corresponding verification transcript under the general shu$le protowI described above and returns T (H, H; G, C) and a (which is equal to cs, known only to the registrant), for each index 1 <_ j _< k. Additionally, in block 6I6, the registrant generates a PKI tertificate request with certain random identifying information. The random identifying information may be any user ID the registrant chooses to employ that cannot be used to identify the registrant_ Under block 616, the registrant also safely stores a corresponding private key based on this request (which differs from the private key s~).
In block 618, the registration server checks the shuffle . verification transcript and checks that h j = G'. If both of these checks pass, then the reBistzation server sets H ~ H' minus the one public key used by the registrant for certification (h~ ), G ~ C and k = k 1. For auditing purposes, the registration server in block 618 also stores the registrant's verification transcript (i.e., T(H, H; G, C)). The registration server also digitally signs the certificate request R to create a PKI

SUBSTITUTE SHEET (RULE ~6) wo omas~ rcrNSoiro9sso certificate that is rebmrned to the registrant. The routine then is ready for the next registrant's xequest Referring to Figure ?, a routine 700 shows the second variant described above for anonymous certificate distribution. The routine 700 is similar to the routine 600. Block ?04 is similar to block 604, except that the set H iacdudes public/private key Pairs, and may be a pmper subset. Similarly, block 710 is similar to black 610, as shown in ' Figure 7.
After receiving a request, the registration server in block 714 retrieves a set H of public key pairs. Under an alternative embodiment, the registration server retrieves only a subset that includes the registrant's public key. 1a block 716, the registrant selects a subset of the public key gains M and sets M' = H M. The registrant computes a ahu~le 15f of M and a coaespondiag verification traasaipt (T (M, H: g.
G7), and generates a zero-Imowledge proof, P that the registrant lmows a secret exponent s as shown in Figure 7. Additionally, the registrant generates pKI certificate request with random identifying information and stores the private key, as described above.
In block 718, ~e rtgistratia~ saver checks the shu$le verificatian transcript and P. If both checks pass, then the registration server sets X (with the public key pair (~f> h j ) removed under equations(18) or (19)) and sets k ~ k-1. The remainder of routine 700 is substantially similar to ti~at of routine 600.
11_ Conclusion One skilled is the art will appreciate that the concepts of the invention can be used in various envimnmcnts ether than the Internet.
For example, the concepts can be used in an electronic mail environment in which electronic avail ballots, transactions, or forms are processed and stored. Im general, a web page or display descritpti~ (e_g., the bulletin board) tray be in HTNB., ?~, or WAP format, email format, or any over format suitable for displaying information (including charaeterlcode SU9STITUTE SHEET (RULE 2B) based formats, bitmapped formats and vector based formats)- Also, various communication channels, Such as local area networks, wide area networks, or point-to-point dial-up connections, may be used instead of the Internet. The various transactions may also be conducted within a single computer environment, rather than in a client/server environment_ Each voter or client computer may comprise any combination of hardware or software that interacts with the server computer or system. These client systems may include television-based systems, Internet appliances and various other consumer products through which transactions can be perfo~cd.
In general, as used heroin, a "link" refers to any resource locator identifying a resource on the network, such as a display description of a voting authority having a site or node on the network. In general, while hardware platforms, such as voter computers, terminals and servers, are described herein, aspects of the invention are equally applicable to nodes on the network having corresponding resource locators to identify such nodes.
Unless the context clearly requires otherwise, throughout the description and the claims, the words 'comprise', 'comprising', and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is to say, in the sense of "including, but not limited to"_ Words using the singular or plural number also include the plural or singular number, respectively. Additionally, the words "herein", "hereuader", and words of similar import, when used in this application, shall refer to this application as a whole and not to any -.particu,>ar portions of this application.
The above description of illustrated embodimcnxs of the invention is not intended to be exhaustive or to limit the invention to the precise form disclosed. While specif c embodiments o~ and examples for, the invention are described herein for illustrative purposes, various equivalent modifications are possible within the scope of the invention, as those skilled in the relevant art will recognize. The teachings of the SUBSTITUTE SHEET (RULE 261 WU 01/7369.1 1'CTNSO1l09S50 invention provided herein can be applied to other encryption applications, not only die eleebronic voting sysbeln described above. For example, the protocol has applications in electronic ~ commerce where both anonymity and auditability are requirements. Examples of this are electronic paymcrrt schemes ("e-cash").
The various embodiments described above can be combined to provide further embodiments. All of the above references end U.S.
pate~ats and applications are incorporated herein by reference. Aspects of the invention can be modified, if necessary, to employ the systems, functions and concepts of the various patents artd applications described above to provide yet further embodiments of the invention.
These and other changes can be made to the invention in light of the above detailed description. 1n general, in the following claims, the terms used should not be construed to limit the invention to the specific embodiments disclosed in the specification and the claims, but should be construed to include all encryption systems and methods that operate ands the claims to provide data security. Accordingly, the invention is not limited by the disclosure, bnt instead the scope of the invention is to be determined entirely by th~claims.
While certain aspects of the invention are presented below is certain claim forms, the inventor contemplates the various aspects of the invention in any number of claim forms. For example, while only one aspect of the invention is recited as embodied in a computer-readable medium, other aspects may likewise be esabodied in computer-readable medium. Accordingly, the inventor reserves the right to add additional claims after filing the application to pursue such additional claim fo~as for other aspects of the invention.

SUBSTITUTE SHEET (RULE 26)

Claims (20)

1. A computer system for receiving a sequence of elements, comprising:
a server computer coupled to a computer network and configured to:
receive a sequence of electronic data elements representing individual data files, apply a cryptographic transformation using at least a secret key to anonymously permute the sequence of electronic data elements and produce a shuffled sequence of electronic data elements, wherein the server computer knows a correspondence between the shuffled sequence of electronic data elements and the sequence of electronic data elements, wherein the server computer is further configured to:
generate a proof of correctness for the permutation based on a proof that the product of unencrypted values of elements of a first sequence of encrypted data elements is equal to the product of unencrypted values of elements of a second sequence of encrypted data elements.

2. The system of claim 1 wherein the received sequence of electronic data elements are encrypted using Z p or elliptic curve groups using a key unknown to the server computer, and wherein the server computer is further configured to:
receive a series of randomly generated values e i from a verifier computer;
and generate the proof of correctness as an non-interactive proof based at least in part on at least some of the randomly generated values e i.

3. The system of claim 1 wherein the server computer is further configured for:
receiving a plurality of public keys from a corresponding plurality of individuals, wherein each of the plurality of individuals have a private key corresponding to one of the plurality of public keys;
receiving a request for a certificate from one of the plurality of individuals having a one private key;
providing at least a subset of the plurality of public keys to the requesting individual;
receiving a shuffle of the plurality of public keys and a non-interactive proof of correctness for the permutation based on a proof that the product of unencrypted values of elements of a first sequence of encrypted data elements is equal to the product of unencrypted values of elements of a second sequence of encrypted data elements;
checking the proof of correctness;
issuing a certificate to the one individual; and reducing the plurality of public keys by the one public key.

4. The system of claim 1 wherein the sequence of electronic elements are public keys, and wherein the server is further configured to check, in response to a request from an individual, that the individual has a secret key uniquely and mathematically related to a one of the public keys; and if so, issue a certificate to the one individual.

5. The system of claim 1 wherein the sequence of electronic data elements is a sequence of ballot choices under an electronic election.

6. The system of claim 1 wherein the proof of correctness proves that given the sequence of electronic data elements and the produced shuffled sequence of electronic data elements, there exists a permutation such that for every decrypted element in the sequence of electronic data elements there exists a corresponding permuted decrypted element in the produced shuffled sequence of electronic data elements.

7. A computer-readable medium whose contents store a sequence of electronic data elements and associated data, wherein the sequence of electronic data elements are processed by a computer-implemented method for a shuffling of the sequence of electronic data elements, the method comprising:
receiving the sequence of electronic data elements;
applying a secret, one-way cryptographic transformation using at least a first secret key to anonymously permute the sequence of electronic data elements and producing a first shuffled sequence of electronic data elements; and generating a proof of correctness for the permutation based on a proof that the product of unencrypted values of elements of a first sequence of encrypted data elements is equal to the product of unencrypted values of elements of a second sequence of encrypted data elements.

8. The computer-readable medium of claim 7 wherein the received sequence of electronic data elements are encrypted with an underlying mathematical group being a ring of integers having a modulus integer value p(Z p).

9. The computer-readable medium of claim 7 wherein the computer-readable medium is logical node in a computer network receiving the sequence of electronic data elements and the proof of correctness.

10. The computer-readable medium of claim 7 wherein the computer-readable medium is computer-readable disk.

11. The computer-readable medium of claim 7 wherein the computer-readable medium is a data transmission medium transmitting a generated data signal containing the sequence of electronic data elements and the proof of correctness.

12. The computer-readable medium of claim 7 wherein the computer-readable medium is a memory of a computer system.

13. The computer-readable medium of claim 7 wherein the computer-readable medium is an Internet connection link to a voting authority server computer.

14. The computer-readable medium of claim 7 wherein the electronic data elements include at least public keys or digital certificates associated with public keys.

15. The computer-readable medium of claim 7 wherein the sequence of electronic data elements are electronic ballots or electronic ballot choices.

16. A computer-implemented method for performing a shuffling of a sequence of electronic data elements, comprising:

providing a request from a computer associated with one private key corresponding to one public key of multiple public keys, wherein each of the multiple public keys corresponds to one of multiple private keys;
receiving a shuffled set of at least some of the multiple public keys; and producing a new shuffled set of the multiple public keys and a proof of correctness for the shuffling, wherein the proof of correctness is based on a proof that the product of unencrypted values of elements of a first sequence of encrypted data elements is equal to the product of unencrypted values of elements of a second sequence of encrypted data elements.

17. The method of claim 16, further comprising:
providing a file; and producing the proof of correctness for the new shuffled set of public keys based on a non-interactive proof that the product of unencrypted values of elements of the first sequence of encrypted data elements is equal to the product of unencrypted values of elements of a second sequence of encrypted data elements.

18. The computer-readable medium of claim 7 wherein the received sequence of electronic data elements are encrypted with an underlying elliptic curve group.

19. The computer-readable medium of claim 7 wherein the proof of correctness is a non-interactive proof of correctness.

20. The system of claim 1 wherein the proof of correctness for the permutation is based on a proof that one polynomial defined by a first sequence of encrypted linear factors is equal to a constant multiple of a second polynomial defined by a second sequence of encrypted linear factors.