The October 13th Galois Tech Talk will be delivered by Brian Huffman, titled “Constructing a Universal Domain for Reasoning About Haskell Datatypes.”

• Date: Tuesday, October 13th, 2009
• Time: 10:30am – 11:30am
• Location: Galois, Inc.421 SW 6th Ave. Suite 300(3rd floor of the Commonwealth Building)Portland, OR 97204

Abstract: Existing theorem prover tools do not adequately support reasoning about general recursive datatypes. Better support for such datatypes would facilitate reasoning about a wide variety of real-world programs, including those written in continuation-passing style, that are beyond the scope of current tools.This paper introduces a new formalization of a universal domain that is suitable for modeling general recursive datatypes. The construction is purely definitional, introducing no new axioms. Defining recursive types in terms of this universal domain will allow a theorem prover to derive strong reasoning principles, with soundness ensured by construction.Bio: Brian Huffman is a PhD student in Computer Science at Portland State University, working with advisor John Matthews. He studies formal reasoning with the Isabelle theorem prover, specializing in formalized mathematics and semantics of functional languages. He is currently the maintainer of Isabelle/HOLCF, a logic for domain theory.

Galois has been holding weekly technical seminars for several years on topics from functional programming, formal methods, compiler and language design, to cryptography, and operating system construction, with talks by many figures from the programming language and formal methods communities. The talks are open and free. An RSVP is not required, but feel free to contact the organizer with questions and comments.

The September 18th Galois Tech Talk will be delivered by Trent Jaeger, titled “Building Systems That Enforce Measurable Security Goals.”[Note the non-standard Friday date; instead of the usual Tuesday slot!]

• Date: Friday, September 18th, 2009
• Time: 10:30am – 11:30am
• Location: Galois, Inc.421 SW 6th Ave. Suite 300(3rd floor of the Commonwealth Building)Portland, OR 97204

The slides for this talk are now available.Abstract: In this talk, I will argue for an approach for building and deploying systems that enforce measurable security goals. Historically, the security community has developed “ideal” goals for security, but conventional systems are not built to satisfy such goals, leading to vulnerabilities. However, we find that building conventional systems to ideal security goals is not a practical option. Ideal security requires heavyweight tasks, such as complete formal assurance, and conventional systems depend on security enforcement in too many programs to make assurance cost-effective. As an alternative, we propose an approach where we use ideal goals as a means to gain a comprehensive understanding of which programs we depend upon for security enforcement and the risks that result from such enforcement. The result is a model that enables comprehensive evaluation of security goals and treatment of risks, once identified. In this talk, I will discuss the motivation for our approach in the development of a practical integrity model, called CW-Lite integrity. Then, I will describe two further experiments. The first examines whether user-level processes can be automatically deployed in a manner in which correct enforcement of system policy can be verified. The second examines whether virtual machine systems can be deployed in a manner in which integrity goals can be determined and verified. In these experiments, we leverage the mandatory access control enforcement of the Linux and Xen, although the talk will focus on the conceptual problems in obtaining a comprehensive view of security in conventional systems. The result of these experiments is that by making security goals measurable in conventional systems a comprehensive view of security can be obtained that enables the solution of key problems in building and deploying secure systems.Bio:Trent Jaeger received his M.S.E. and Ph.D degrees in computer science and engineering from the University of Michigan, Ann Arbor in 1993 and 1997, respectively. Trent joined the Computer Science and Engineering department at Penn State in 2005. He is co-director of the Systems and Internet Infrastructure Security (SIIS) Lab at Penn State. Prior to joining Penn State, he was a research staff member at IBM Thomas J. Watson Research Center for nine years.His research/teaching interests are in the areas of computer security, operating systems, security policies, and source code analysis for security. Trent has been active in the Linux community for several years, particularly in contributing code and tools for the Linux Security Modules (LSM) framework (in Linux 2.6) and for integrating the SELinux/LSM with IPsec (called Labeled IPsec, available in Linux 2.6.18 and above). Trent also has active interests in virtual machine systems (mostly Xen) and trusted computing hardware (Linux Integrity Measurement Architecture and its successor PRIMA).He is also active in the security research community at large, having served on the program committees of many of the major security conferences, including the IEEE Symposium on Security and Privacy, USENIX Security Symposium, ACM Conference on Computer and Communications Security, European Symposium on Research in Computer Security, Annual Computer Security Applications Conference, ISOC Network and Distributed Systems Symposium. Trent has been Program Chair of the ACM Symposium on Access Control Models and Technologies and the ACM Conference on Computer and Communications Security (Industry Track). He is the current Program Chair for the 2007 USENIX Workshop on Hot Topics in Security. He has over 60 refereed publications, and is the holder of several U.S. patents.

Galois has been holding weekly technical seminars for several years on topics from functional programming, formal methods, compiler and language design, to cryptography, and operating system construction, with talks by many figures from the programming language and formal methods communities. The talks are open and free. An RSVP is not required, but feel free to contact the organizer with questions and comments.

Substitution ciphers are one of the oldest encryption methods, dating back to at least the 15th century. In a substitution cipher, each character in the plain-text is simply “substituted” according to a predefined map. Decryption is simply the substitution in the reverse direction. Wikipedia has a nice description of these ciphers. Obviously, you wouldn’t want your bank to use such a cipher when executing your web-based transactions! But they are fun to play around, especially when entertaining kids in hot summer days. In this post, we’ll see how to code simple substitution ciphers in Cryptol, and go a step further and actually prove that our implementation is correct.

Preliminaries

The simplest form of substitution ciphers use a permutation of the input alphabet. That is, each letter in the input alphabet gets mapped to another in the same  alphabet. (Strictly speaking, input and output alphabets need not be the same, but nothing essential changes by making that assumption.) For instance, you might decide that your substitution will map ‘a’ to ‘q’, and ‘b’ to ‘d’, …, etc., making sure no two letters are mapped to the same target. Once this mapping is agreed on, all you have to do to encrypt a given message is to map each character to the corresponding element according to your predefined mapping rules.Here’s our Cryptol encoding of these ciphers. First, some preliminary declarations:

type Char = [8];type String(l) = [l]Char;type Table(n) = [n](Char, Char);

We’ll simply assume that the input consist of “characters,” each of which will be 8-bit quantities (i.e., numbers from 0 to 255). We will simply use ASCII encoding for normal English characters. This is captured by the Char type declaration above, which simply gives a convenient name for 8-bit wide words. The second type declaration captures sized-strings: For any given size l, the type String(l) represents a sequence of length l, containing 8-bit words. For instance, String(16) is the type of all sequences of length 16, containing numbers from 0 to 255 as elements. Finally a Table of size n is simply n-pairings of characters that form a substitution. Here’s the example table we will use:

cipherTable : Table(28);cipherTable = [| (x, y) || x <- plain || y <- cipher |]where { plain = "abcdefghijklmnopqrstuvwxyz .";cipher = "oebxa.cdf hijklmnzpqtuvwrygs"};

Note that our table has 28 entries (the lower-case English alphabet, plus space and the dot). A simple Cryptol sequence-comprehension succinctly zips the sequences up, forming our example table.

Consider the following multiplication algorithm, coded in Mostek 6502 Assembler:

 LDX #8 ; 1; load X immediate with the 8LDA #0 ; 2; load A immediate with the 0CLC ; 3; set C to 0LOOP ROR F1 ; 4; rotate F1 right circular through CBCC ZCOEF ; 5; branch to ZCOEF if C = 0CLC ; 6; set C to 0ADC F2 ; 7; set A to A+F2+C and C to the carryZCOEF ROR A ; 8; rotate A right circular through CROR LOW ; 9; rotate LOW right circular through CDEX ;10; set X to X-1BNE LOOP ;11; branch to LOOP if Z = 0

This program comes from Wilfred Legato’s paper “A Weakest Precondition Model for Assembly Language Programs.” It multiplies the contents of the memory locations F1 and F2; each of which is 8-bits wide. The result is stored in the accumulator register A and the memory location LOW, each of which is, again, 8-bits. It holds that:

 F1 * F2 = 256 * A + LOW

when the algorithm terminates, correctly handling the overflow. It is worth spending a moment or two pondering how this algorithm works; it is not at all obvious how the multiplication is done!Legato’s challenge  (as  referred to in ACL2 circles) is to prove a deep-embedding of Legato’s algorithm correct with respect to a Mostek simulator coded in ACL2. We do not attempt to solve  Legato’s challenge in Cryptol. We are merely interested in coding and proving that Legato’s multiplier is correct in Cryptol. Our interest stems from the fact that Legato’s algorithm is a truly interesting multiplier on its own right, and we would like to make sure that a straightforward encoding of it in Cryptol can be proven correct automatically by Cryptol’s verification tools. And of course, it’s just too hard to pass up on the opportunity to  pay respect to the Mostek chip that powered the Commodore 64‘s and Atari 800XL‘s of our childhood.

A shallow embedding

The Cryptol solution to Legato’s problem will be a fairly shallow encoding of the multiplier, together with an automated proof of correctness. We choose to do a shallow encoding here since it allows us to focus on the multiplication algorithm itself, as opposed to the particulars of the underlying Mostek chip. Theorem proving based solutions (such as those given by ACL2 folks) will rightly pursue a deeper embedding of the algorithm and the Mostek architecture in general. Cryptol is not particularly suitable for deep embeddings. Representing Mostek assembly instructions directly as Cryptol functions is a much simpler and straightforward choice.Looking at Legato’s multiplier above, we will represent each instruction (from 1 to 11) as a simple state transformer, taking a simplified representation of the Mostek machine state as input and delivering a new one. We will only represent parts of the state that matter for our problem. The following Cryptol type declaration succinctly captures what we need:

 type Mostek = ( [8] // F1, [8] // F2, [8] // A, [8] // X, [8] // LOW, Bit // C (Carry), Bit // Z (Zero));

Using this state representation, each instruction in the program can be modeled as a  state transformer:

 type Instruction = Mostek -> Mostek;

This takes care of the data-flow aspect of the embedding; but the question of how to model control-flow remains. We will simply use the host-language’s control-flow features, using the quintessential functional idiom: by calling functions! This is actually easier done than said, and here’s our embedding of the first instruction of the program:

 // step1: LDX #8; load X immediate with the integer 8.step1 : Instruction;step1 (f1, f2, a, _, l, c, z) =step2 (f1, f2, a, 8, l, c, z);

Let’s spend a minute explaining this in detail. The first step in the program loads the register X with the immediate value 8. Using our state-transformer model, our step1 function will receive a Mostek state (consisting of the “current” values of F1, F2, A, X, LOW, CARRY, and ZERO). The “effect” of this instruction is to put the value 8 into the register X, leaving everything else the same. Once this is done, the control goes to the next instruction, which we model by calling the function step2 (which is yet to be defined).In this fashion, we can shallowly embed all the instructions in Legato’s multiplier, using Cryptol’s native functions and control-flow features. Of course, this is hardly a new idea, being the essence of the whole domain-specific embedded language saga: Using a rich host-language to “fake” other languages.Following the recipe set by step1, it is easy to model the next two instructions:

 // step2: LDA #0; load A immediate with the integer 0.step2 : Instruction;step2 (f1, f2, _, x, l, c, z) =step3 (f1, f2, 0, x, l, c, z);// step3: CLC; set C to 0 (Note the use of Bit False here)step3 : Instruction;step3 (f1, f2, a, x, l, _, z) =step4 (f1, f2, a, x, l, False, z);

Step 4 is equally easy in terms of control flow, but is tricky in terms of operation. After some head-scratching, one figures out that the term “rotate F1 right circular through C” means put the right-most bit of F1 in C, and put C in the first position of F1. A bizarre thing to do indeed, but that’s the beauty of Legato’s multiplier. The Cryptol translation is almost literal:

// step4: LOOP ROR F1; rotate F1 right circular through C.step4 : Instruction;step4 (f1, f2, a, x, l, c, z) =step5 (f1', f2, a, x, l, b0, z)where {[b0 b1 b2 b3 b4 b5 b6 b7] = f1;f1' = [b1 b2 b3 b4 b5 b6 b7 c];};

The use of pattern matching in getting the bits out of f1, and the construction of the new value of f1 is idiomatic Cryptol. There’s one little catch though: Apparently Mostek was a big-endian machine, having a most-significant-bit-first representation. Cryptol is little-endian. So, instead of rotating the bits to right, we  rotate them left.The fifth instruction is the first time where we use Cryptol’s control-flow to model the Mostek jump instruction:

 // step5 : BCC ZCOEF; branch to ZCOEF if C = 0.// ZCOEF is step8 in our encodingstep5 (f1, f2, a, x, l, c, z)= if c then step6 (f1, f2, a, x, l, c, z)else step8 (f1, f2, a, x, l, c, z);

In this case, we simply receive a state, and depending on the value of the carry bit (C), we either go to the next step (i.e., no jump); or go to the ZCOEF instruction, which is going to be step-8 in our model. Easy as pie!Step-6 is a replica of Step-3, clearing the carry bit:

 // step6: CLC; set C to 0step6 (f1, f2, a, x, l, _, z) =step7 (f1, f2, a, x, l, False, z);

Step-7 is the most compute intensive part of the algorithm. The Cryptol encoding is a bit complicated due to the need to determine if there was a carry in the addition. Since all Cryptol arithmetic is modular, we are forced to do the computation at an extended bit-size. Otherwise, the modeling of the ADC instruction is quite straightforward:

 // step7: ADC F2; set A to A+F2+C and C to the carry.step7 (f1, f2, a, x, l, c, z) =step8 (f1, f2, a', x, l, c', z')where {// 8-bit "modular" resulta' = a + f2 + (if c then (1:[8]) else (0:[8]));// Was there a carry? Check that "real"// result is larger than 255a'Large : [9];a'Large =(a # zero) // extend a by adding zero bits+ (f2 # zero) // same for f2+ (if c then (1:[9]) else (0:[9]));c' = a'Large > (255:[9]);// set the zero flagz' = a' == 0;};

The Cryptol idiom x # zero simply represents the value x extended on the right with 0 bits. (Remember that Cryptol is little-endian, hence the addition of zero bits on the right does not change the value.) Due to the polymorphic type of the value zero, the result has any number of bits larger than equal to the original bit-size of x. (Since we only need 9-bits i
n this case, we could have coded the same via the expression x # [False], but the former expression is more idiomatic Cryptol.)Steps 8 and 9 are similar to Step-4, using A and LOW instead of F1, respectively:

 // step8 : ZCOEF ROR A; rotate A right circular through C.step8 : Instruction;step8 (f1, f2, a, x, l, c, z) =step9 (f1, f2, a', x, l, a0, z)where {[a0 a1 a2 a3 a4 a5 a6 a7] = a;a' = [a1 a2 a3 a4 a5 a6 a7 c];};// step9 : ROR LOW; rotate LOW right circular through C.step9 : Instruction;step9 (f1, f2, a, x, l, c, z) =step10 (f1, f2, a, x, l', l0, z)where {[l0 l1 l2 l3 l4 l5 l6 l7] = l;l' = [l1 l2 l3 l4 l5 l6 l7 c];};

Step-10 simply decrements X, setting the ZERO flag appropriately:

 // step10: DEX; set X to X-1step10 : Instruction;step10 (f1, f2, a, x, l, c, z) =step11 (f1, f2, a, x', l, c, x'==0)where x' = x-1;

Finally, step-11 either jumps back to the top of the loop (step-4), or finishes the algorithm:

 // step11: BNE LOOP; branch to LOOP if Z = 0.// LOOP is step4 in our encodingstep11 : Instruction;step11 (f1, f2, a, x, l, c, z)= if zthen (f1, f2, a, x, l, c, z) // done!else step4 (f1, f2, a, x, l, c, z);

From a control-flow perspective, we indicate the end of the algorithm by simply returning the final Mostek state. It is worthwile at this point to go through the Cryptol embeddings of the instructions to see how they match-up to the Mostek assembly given by Legato.

Extracting the multiplier

Having coded Legato’s multiplier as a sequence of state transformers, we can simply call the function step1 to use it with an appropriate state. The following helper function simplifies this task for us, by loading the registers F1 and F2, and extracting the high and low bits at the end:

legato : ([8], [8], Mostek) -> ([8], [8]);legato (f1, f2, st) = (hi, lo)where {// get the relevant parts// to construct the starting state(_, _, A, X, LOW, C, Z) = st;// Run legato multiplier;// final A is hi; and final LOW is low(_, _, hi, _, lo, _, _) =step1 (f1, f2, A, X, LOW, C, Z);};

Note that legato still takes the starting machine state st as an argument. Legato’s claim (which we will shortly prove) is that the algorithm works correctly no matter what the initial state is, hence it is important to be explicit about the starting state.To see legato in action, let’s just run it on a simple input:

 legato> legato (12, 93, (9, 42, 3, 8, 1, False, True))(4, 92)

where I just made up the initial state by plugging in some random values. If Legato is right, then it must be the case that

 12 * 93 = 256 * 4 + 92

correctly computing the high and low bytes. And voila! Both sides equal 1116. Magic!

Correctness

If you do believe in magic,  you can stop reading now. But I suspect most readers of the Galois blog will be looking for something more concrete. Surely, we must be able to give a better argument than claiming witchcraft for the correctness of our implementation.Let us first formally capture what we mean by “correct,” by writing a Cryptol theorem that expresses our intuitive expectation:

theoremlegatoIsCorrect: {x y st}. x' * y' == 256 * hi' + lo'where { (hi, lo) = legato (x, y, st);hi', lo', x', y' : [16];hi' = hi # zero;lo' = lo # zero;x' = x # zero;y' = y # zero};

Here’s the English reading of this theorem: “For all values of x, y, and st, if we run legato on these values and get the results hi and lo, then, it’ll be the case that x * y = 256 * hi + lo.” The only caveat is that we have to do arithmetic operations over 16 bit values (instead of 8), to make sure the theorem statement correctly captures the intended mathematical meaning. (Recall that all Cryptol arithmetic is modular with respect to the bit-size involved.) Hence, we simply add extra zero‘s at the end to enlarge the arguments to 16 bits. Note that, we do not have to assert that the value of lo is at most 255; this is automatically guaranteed by the fact that it is an 8-bit value. Cryptol’s bit-precise type system saves the day!

Verification

Here’s what happens when I run cryptol on the file containing the above theorem:

$ cryptol legato.cryCryptol version 1.8.5, Copyright (C) 2004-2009 Galois, Inc.www.cryptol.netType :? for helpLoading "legato.cry".. Checking types.. Processing.. Done!*** Auto quickchecking 1 theorem.*** Checking "legatoIsCorrect" ["legato.cry", line 147, col 1]Checking case 100 of 100 (100.00%)100 tests passed OK[Coverage: 3.47e-14%. (100/288230376151711744)]

When Cryptol sees a theorem declaration in a loaded file, it automatically performs a quick-check run to provide feedback on its validity. In this case, Cryptol automatically created 100 random test values for the theorem and checked that each one of them satisfied the statement. This is a quick way of getting feedback on the correctness of theorems, courtesy of Cryptol at no additonal cost to the user!While the quick-check run is promising, the coverage info indicates that we’ve barely scratched the surface. The entire state space in this case has 58 bits (8 each for x and y, plus the starting arbitrary state of the Mostek machine costing us an extra 42 bits; for a total of 58). The total number of possible inputs is, therefore, 2⁵⁸ or 288230376151711744. This is a huge number: If you had a computer that run 1-billion (10⁹) test cases every second, it’d still take you over 9 years to go through all possible inputs!Of course, we can do better. Cryptol’s theorem proving environment uses modern equivalence-checkers to prove such theorems automatically, at the push of a (virtual) button:

 legato> :prove legatoIsCorrectQ.E.D.

And there, we’ve proved that our implementation of Legato’s multiplier is indeed correct for all possible inputs! (The above proof takes about 2.5 minutes to complete on my 3-year old MacBook Pro, using abc as the underlying equivalence checker in Cryptol’s symbolic mode. I should also note that the symbolic mode is only available in the full Cryptol release, for which free licenses are available.)

Closing thoughts

I must emphasize that we are not advocating Cryptol as a platform for doing proofs of algorithm correctness. Modern theorem provers such as ACL2, Coq, or Isabelle are the leading tools in this regard. (In particular, the logic behind Cryptol’s automated theorem prover is much less expressive, for starters.) Where Cryptol shines is in its restricted attention to bit-vectors and data-flow algorithms (cryptography being a prime application area), and it turns out that automated equivalence-checking based techniques do perform rather well for such problems. Our shallow embedding of Legato’s multiplier and the automated proof-of-correctness is a case in point.There is one more important point to make. While push-button provers are indispensable in industrial practice, the final Q.E.D. you get from an interactive theorem prover such as ACL2 or Isabelle is much more satisfactory. For instance, we can hardly claim that the above proof increased our understanding of Legato’s algorithm in any sense, it just made us really believe it. I’m willing to bet that anyone who goes through a similar proof in ACL2 or Isabelle would have a much higher chance of having their “aha!” moment, where everything just sinks in…On the practical side, however, nothing beats the fully-automated Q.E.D., especially when your boss is breathing down your neck!

Download

The Cryptol file containing Legato’s multiplier and the correctness theorem is here. The Cryptol toolset licenses are freely available at www.cryptol.net.

The June 30th Galois Tech Talk will be delivered by Brian Huffman, titled “Verifying Stream Fusion with Isabelle/HOLCF”

• Date: Tuesday, June 30th, 2009
• Time: 10:30am – 11:30am
• Location: Galois, Inc.421 SW 6th Ave. Suite 300(3rd floor of the Commonwealth Building)Portland, OR 97204

Slides from the talk.Abstract: Stream fusion is a system for removing intermediate data structures from Haskell programs that manipulate lists.  Formal verification of such libraries requires very precise modeling in a theorem prover, to avoid strictness-related bugs.In this talk I will present a formalization of the stream fusion library in Isabelle/HOLCF, a theorem proving environment designed especially for reasoning about functional programs.  I will show how to prove the correctness of various stream functions using “fixed-point induction”, a powerful reasoning principle for general recursive functions.Bio: Brian Huffman is a PhD student in Computer Science at Portland State  University, working with advisor John Matthews. He studies formal reasoning with  the Isabelle theorem prover, specializing in formalized mathematics and semantics of functional languages. He is currently the maintainer of Isabelle/HOLCF, a logic for domain theory.

Galois has been holding weekly technical seminars for several years on topics from functional programming, formal methods, compiler and language design, to cryptography, and operating system construction, with talks by many figures from the programming language and formal methods communities. The talks are open and free. An RSVP is not required, but feel free to contact the organizer with questions and comments.

The June 23rd Galois Tech Talk will be delivered by John Launchbury, titled “A Newbie’s Exploration of Separation Logic.”

• Date: Tuesday, June 23rd, 2009
• Time: 10:30am – 11:30am
• Location: Galois, Inc.421 SW 6th Ave. Suite 300(3rd floor of the Commonwealth Building)Portland, OR 97204

Abstract: I was just privileged to be in a Separation Logic Tutorial, given by its inventor, John Reynolds. Separation logic allows descriptions of storage and other resources concisely, providing a novel system for reasoning about imperative programs with shared mutable data structures. Recent years have seen a flurry of activity in Separation Logic, extending it to apply from shared-variable concurrency to permission based access control mechanisms. In this informal chalk-talk I will introduce the basics of Separation Logic, providing an overview of the fundamental notions of proof techniques.Bio: John Launchbury is the CTO of Galois, Inc. Prior to founding Galois in 1999, John conducted research and instructed in Computer Science and Engineering at the Oregon Graduate Institute School of Science and Engineering at OHSU. John received First Class Honors in Mathematics from Oxford University in 1985. He holds a Ph.D. in Computing Science from University of Glasgow and won the British Computer Society’s distinguished dissertation prize.

Galois has been holding weekly technical seminars for several years on topics from functional programming, formal methods, compiler and language design, to cryptography, and operating system construction, with talks by many figures from the programming language and formal methods communities. The talks are open and free. An RSVP is not required, but feel free to contact the organizer with questions and comments.

We have a new position paper on the use of EDSLs (LwDSLs) for verification and validation of unmanned vehicle avionics, written jointly with John van Enk of DornerWorks, recently presented at a mixed-criticality architecture conference. (Download) :: PDF

Lee Pike, Don Stewart, John Van EnkCPS Week 2009 Workshop on Mixed CriticalityRoadmap to Evolving UAV Certification

We outline a new approach to the verification and validation (V & V) of safety-critical avionics based on the use of executable lightweight domain specific languages – domain-specific languages hosted directly in an existing high-level programming language. We provide examples of LwDSLs used in industry today, and then we describe the advantages of LwDSLs in V & V. We argue the approach promises substantial automation and cost-reduction in V & V.

I am on the program committees for two upcoming formal methods conferences: Formal Methods in Computer-Aided Design (FMCAD), the preeminent conference on formal methods in hardware and systems, and Automated Formal Methods (AFM), a workshop on the application, usage, and extension of formal methods tools, particularly focusing on SRI’s tool suite (including a theorem prover, model-checkers, and SMT solver).Please consider submitting papers!  The deadline for FMCAD is May 22 (with abstracts due May 15); the deadline for AFM is April 30.  FMCAD will occur in Austin, Texas November 15-18, and AFM will be colocated with CAV in Grenoble, France.

Cryptol is a language tailored for cryptographic algorithms. Sudoku is a popular puzzle the reader  is no-doubt already familiar with. We will offer no deep reason why anyone should try to solve Sudoku in Cryptol; other than the very fact that it’d be a shame if we couldn’t!Needless to say, Cryptol has not been designed for encoding search algorithms. Nonetheless, some of the features of Cryptol and its associated toolset make it extremely suitable for expressing certain constraint satisfaction problems very concisely; and Sudoku very nicely falls into this category.

Representing the board

A Sudoku board can be represented in a variety of ways. We will pick the simplest: A sequence of 9 rows, each of which has 9 elements storing the digits. Each digit will require 4 bits; since they range from 1 to 9. So, a good Cryptol type for a board is:

  [9][9][4]

In Cryptol-speak, this type simply represents a sequence of precisely 9 elements, each of which is a sequence of 9 elements themselves, each of which are 4-bit words. (Technically, the type [4] also represents a sequence of precisely 4 elements, each of which are bits. But it’s easier to read that as 4-bit words. The type [4] and [4]Bit are synonymous in Cryptol, and can be used interchangeably in all contexts.)

Accurate and reliable elections are a critical component of an effective democracy. However, completely secure and trustworthy voting procedures are difficult to design, and no perfect solutions are known. Ideally, a trustworthy voting system should guarantee both verifiability (the ability to prove that the counted vote matches the submitted ballots) and privacy (the inability to link the contents of a vote with the voter who cast it).These guarantees may now be achievable. Many researchers have proposed voting protocols that achieve verifiability and privacy in theory, and a few do so under assumptions that are satisfied by current election practices. Most of the protocols involve posting an encrypted version of the contents of every ballot in some public place (likely a web site), and depend on the properties of cryptographic operations to achieve privacy while allowing anyone to verify the final tally. Now that practical, secure voting protocols exist, the time has come to bring them into use. One existing solution that comes close to achieving these goals while retaining compatibility with current voting practices is the Scantegrity II system. It has the advantage that it can operate under current US election conditions, without requiring any modification to existing optical ballot scanners, and with very little change to the individual voting process. However, the software used in this system is only a prototype, with a number of shortcomings. Voter privacy depends on ability of a computer system to keep a key database completely secret, and accurate vote counting depends on the correct implementation of complex cryptographic algorithms. The software is tens of thousands of lines of code, and as with any other software of that size, many bugs certainly exist. We believe that the importance of trustworthy election results and the past lack of success in creating reliable solutions warrants a new approach to the design of voting systems. In particular, we advocate a class of techniques known as formal methods that allow us to make precise mathematical assertions about how software should behave, and determine whether it satisfies those assertions. Government agencies within the Department of Defense make use of formal methods to ensure the reliability of important computer systems, and the draft update to the development standards used by the Federal Aviation Administration, DO178C, includes provisions for the use of formal methods. Voting systems deserve similar care.