Abstract—This paper describes an automatic method for proving equivalence of implementations of block ciphers (and similar cryptographic algorithms). The method can compare two object code implementations or compare object code to a formal, mathematical specification. In either case it proves that the computations being compared are bit-for-bit equivalent. The method has two steps. First the computations are represented as large mathematical terms. Then the two terms are proved equivalent using a phased approach that includes domain-specific optimizations for block ciphers and relies on a careful choice of both word-level and bit-level simplifications. The verification also relies on STP [5], a SAT-based decision procedure for bit-vectors and arrays. The method has been applied to verify real, widely-used Java code from Sun Microsystems and the open source Bouncy Castle project. It has been applied to implementations of the block ciphers AES, DES, Triple DES (3DES), Blowfish, RC2, RC6, and Skipjack as well as applications of the cryptographic hash functions SHA-1 and MD5 on fixedlength messages. I.

We show how the binary encoding and decoding of typed data and typed programs can be understood, programmed, and verified with the help of question-answer games. The encoding of a value is determined by the yes/no answers to a sequence of questions about that value; conversely, decoding is the interpretation of binary data as answers to the same question scheme. We introduce a general framework for writing and verifying gamebased codecs. We present games for structured, recursive, polymorphic, and indexed types, building up to a representation of well-typed terms in the simply-typed λ-calculus. The framework makes novel use of isomorphisms between types in the definition of games. The definition of isomorphisms together with additional simple properties make it easy to prove that codecs derived from games never encode two distinct values using the same code, never decode two codes to the same value, and interpret any bit sequence as a valid code for a value or as a prefix of a valid code.

Cryptol is programming a language designed for specifying and programming cryptographic algorithms. In order to meet highassurance requirements, Cryptol comes with a suite of formalmethods based tools allowing users to perform various program verification tasks. In the fully automated mode, Cryptol uses modern off-the-shelf SAT and SMT solvers to perform verification in a push-button manner. In the manual mode, Cryptol produces Isabelle/HOL specifications that can be interactively verified using the Isabelle theorem prover. In this paper, we provide an overview of Cryptol’s verification toolset, describing our experiences with building a practical programming environment with dedicated support for formal verification. Categories and Subject Descriptors F.3.1 [Logics and meanings

Sparse matrix formats are typically implemented with low-level imperative programs. The optimized nature of these implementations hides the structural organization of the sparse format and complicates its verification. We define a variable-free functional language (LL) in which even advanced formats can be expressed naturally, as a pipeline-style composition of smaller construction steps. We translate LL programs to Isabelle/HOL and describe a proof system based on parametric predicates for tracking relationship between mathematical vectors and their concrete representations. This proof theory automatically verifies full functional correctness of many formats. We show that it is reusable and extensible to hierarchical sparse formats. Categories and Subject Descriptors D.3.2 [Programming Languages]: Language Classifications—Specialized application languages;