Whitfield Diffie and Martin Hellman startled the computer security world in 1975 with their paper “New Directions in Cryptography”, which introduced public-key cryptography [7]. There is now evidence that three cryptographers at the British Government Communications Headquarters (GCHQ), the British equivalent of the National Security Agency, may have predated this work as well as the discovery of the Rivest-Shamir-Adleman (RSA) public-key encryption algorithm [9]. It appears, however, that the British researchers did not appreciate the significance of their discovery. Since the mid 1970s, cryptography has become big business, a bestseller (Applied Cryptography by Bruce Schneier has sold over two hundred thousand copies), and an extremely active area of research at the intersection of mathematics and computer science. A plethora of new books in the area currently floods the bookstores. This review cannot discuss them all. Instead I will delineate the subareas of this burgeoning field and focus on some of the texts that I believe will be of more interest to Bulletin readers.

The Yamabe problem

Universal coalgebra: a theory of systems

In the semantics of programming, finite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with infinite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages (Milner, 1980; Park, 1981). Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras (Aczel and Mendler, 1989). Thus the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to: coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken as the basic ingredients of a theory called universal coalgebra. Some standard results from universal algebra are reformulated (using the afore mentioned correspondence) and proved for a large class of coalgebras, leading to a series of results on, e.g., the lattices of subcoalgebras and bisimulations, simple coalgebras and coinduction, and a covariety theorem for coalgebras similar to Birkhoff's variety theorem.

Structure and interpretation of computer programs

59.4 T h e C u t Elimination Theorem for Second Order Logic 171 (respectively, v . ) . In particular, (a) guarantees t h a t *"-PIND or c ~ ' ~ I N D , and C ~ ' ~ C R inferences are still valid after the substitution of V for a. However, P ( V ) may fail t o be a proof in t h a t there may be initial sequents of P ( V ) of the form where V={Z')A(Z'). However, sequents of this form are easy t o prove without free cuts. S o we merely tack onto P ( V ) free cut free proofs of these initial sequents and thus obtain a proof Q of I'(V)+A(V). Q is not necessarily free cu t free, as Q may contain free cu ts with principal formula A(3). B u t since V is a first order abstract, every free cut inference in Q has a first order principal formula. Hence, by (b), there is a free cut free proof of I'(V)--+A(V). Theorem 10: (Cut Elimination Theorem). Let P be a firproof o r a VTproof. Then there is a proof P * in the same theory such tha t P * has the same endsequent as P and there are no free cu ts in P * . Furthermore, each principal formula of an induction inference in P * is a substitution instance of a principal formula of an induction inference in P and each principal abstract of a comprehension inference in P * is a substitution instance of a principal abstract of a comprehension inference in P. Hence, for all i 2 0 , if P is a fii(or vi-) proof then so is P*. Proofi We shall modify Takeuti 's exposition on pages 22-29, 112, 143-144 of [28]. T h e reader should have [28] available as he reads the proof. Following [28], we define the grade of a formula A t o be the number of logical symbols in A . T h e level of A is the number of second order quantifiers in A . A mix inference with principal formula A is of the form where l l* and A * are obtained from rI and A by removing all occurrences of A . A mix inference is free iff all of the occurrences of A in A and II are free. Since a mix inference and a cut inference are so similar, i t suffices t o prove Theorem 10 for proofs which use mix inferences instead of cu t inferences. Suppose P is a proof whose last inference is a mix with principal formula A as shown above. Define the distance of a sequent in P t o be the number of inferences separating i t from the endsequent of P. T h e right rank of P is defined t o be the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent ll. T h e left rank of P is the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent 172 Foundations of Second Order Bounded Arithmetic A . T h e rank of P is the sum of the right rank and left rank. I t suffices t o consider P with a single mix inference as the last inference. T h e proof is by ordinal induction on where level(P) and grade(P) are the level and grade of the principal formula of t he final mix of P, and w is the first infinite ordinal. T h u s it suffices t o show tha t if P is a proof with no free mixes except for the final inference of P and if Theorem 10 is satisfied for all proofs P' with ord(P' )<ord(P) , then P satisfies Theorem 10. W e modify the proof of Lemma 5.4 of Takeut i [28]: Case (1): rank(P)=2. Cases (1 .I)-(1.5. ii): Similar t o pages 2 4 2 7 of [28]. Case (1.5.iii): Suppose A=(V+)B(q5) and the last inferences of P are where V is a ~ d ~ ~ a b s t r a c t , and since rank(P)=f the indicated occurrences of (V+)B(+) are the only ones. By Lemma 9(c), we can obtain a free mix free proof of rA,B( V) from the free mix free proof of I'-+A,B(cr). T h u s we have a proof Q such t h a t the only free mix in Q is i ts last inference: where II# and A# are II and A minus all occurrences of B(V). By the induction hypothesis, there is a free mix free proof Q* of ~,II#-A#,A since ord(Q)<ord(P). By adding weak inferences t o the end of Q* we obtain the desired proof P* . Case (1.5. i~) : Suppose A=(34)B(4). This case is very similar t o Case (1.5.iii). Case (2): rank(P) >2. Case (2.1): T h e right rank of P is >l. Cases (2.1.1)-(2.1.9. ii): Similar t o 1281. T h e C u t Elimination Theorem for Second Order Logic 173 Case (2.1.J.iii): Suppose A=(V4)B(4), V is a ~:p~-abstract and the last inferences of P are: where now A and II, b u t not r , may contain occurrences of (Vc$)B(4). T h e cedents TI* and A* are II and A minus all occurrences of (V+)B(c$). Modify the end of P t o obtain a proof P1 which ends as T h e right rank of P1 is one less than the right rank of P so by the induction hypothesis there is a free mix free proof Pg of the endsequent of P I . Now consider the proof which ends T h e right rank of this is one, so by the induction hypothesis and some exchanges and contractions we obtain a free mix free proof of r , n * + A*,A. T h e rest of the cases are similar. Q.E.D. 9.5. C:yb-Defined Functions and A:yb-Defined Predicates. T h e second order theories of Bounded Arithmetic are in many respects analogous t o the first order theories s~~ and T;. One of the most fundamental properties of second order Bounded Arithmetic is t ha t new function and predicate symbols may be introduced into the language of Bounded Arithmetic; under certain conditions, these new function and predicate symbols may be used freely in the principal formulae of induction axioms and comprehension axioms. 174 Foundations of Second Order Bounded Arithmetic Definition: Let R be a second order theory of Bounded Arithmetic. Suppose A(2,y) is a formula with all free variables indicated and tha t Then we say t h a t R can define the function j such t h a t N t= ('d$)A(Z',j($)). T h e ~ t ) ~ d e f i n e d function symbols and the Atpb-defined predicate symbols play the same role in the second order theories of Bounded Arithmetic as the C:-defined function symbols and the A/-defined predicate symbols d o in the first order theories Si and T;'. In particular, the analogues of Theorems 2.2, 2.3 and 2.4 hold for second order Bounded Arithmetic. Definition: Let 7 and ji be new function and predicate symbols. T h e sets C,'tb(7,ji) and ll,'*b(7,j?) are sets of bounded formulae in the language of second order Bounded Arithmetic plus the symbols f' and ji. These sets are defined by counting alternations of second order quantifiers ignoring the first order, bounded quantifiers. Theorem 11: Let R be a second order theory of Bounded Arithmetic. Suppose R can C;lb-define each of the functions f' and can define each of the predicates ji. Let R * be the theory obtained from R by adjoining the new symbols 7 and j3 and their defining axioms. Then, if i > O and B is a C;'lb(7,j?)(or a Il,'lb@,j?)-) formula, then there is a formula B*Ec;'~~ (or II,'pb, respectively) such t h a t R * t B** B. T h e proof of Theorem 11 is similar t o the proofs of Theorems 2.2 and 2.4. Definition: Let R be a theory of Bounded Arithmetic and let 7 be a vector of defined function symbols of R and j3 be a vector of defined predicate symbols. Then ~(7, j i ) is the conservative extension of R obtained by enlarging the language t o include 7 and 3 and including the defining axioms for these symbols. Corollary 12: ( i 2 1): (a) Let f' be a vector of ~ t * ~ d e f i n e d function symbols of U; (respectively, v;) and let ji be a vector of At*b-defined predicate symbol-. of u;' (respectively, v;' ). Then U;@,j3) (respectively, vfl,j?) ) has as theorems the E : ~ ~ ~ , ~ ~ ) P I N D axioms (respectively, the c,"~(~!,~~)-IND axioms). (b) Let 7 be a vector of C:rb-defined function symbols of fi; (respectively, F;' ) and let ji be a vector of A:.b-defined predicate symbols of 6; (respectively, ) Then fi;'(],j?) (respectively, ?i@,j?)) has as theorems the E:I*(~!,~I)-PIND axioms (respectively, the E,'P~@,~~)-IND axioms). (c) Let f' be a vector of C:lb-defined function symbols of 8; (respectively, 8;:) and let j? be a vector of A:rb-defined predicate symbols of 8; (respectively, ) Then 8i(],jI) s9.5 zipb-Defined Functions and airb-Defined Predicates 175 (respectively, ) has as theorems the E:V~-PIND v , a ) axioms (respectively, the c~'~~-IND (7,jJ) axioms) and the i :9b(7,$)-~~ axioms. Corollary 12 tells us t h a t C:~~-defined function symbols and a:1b-defined predicate symbols may be used freely in the principal formulae of induction inferences. Furthermore, if we are working in the theory 82 or 82 we may use such function and predicate symbols freely in principal abstracts of comprehension inferences. T h e next two theorems give an application of ~ : ~ ~ c o m ~ r e h e n s i o n t show t h a t 82 and 8: can define the iteration of defined predicates. I t is an open question whether these theorems hold for the theories U; and v;. Theorem 18: Let ~(a,i?,?) and ~ ( a , b,t,a:,q) be formulae of 8:, where a: is a unary predicate variable. Let t (b , t ) be a term which contains only the free variables b and t. Then the predicate ~ ( a , b,i?,;) which satisfies if b=O and a < t ( b , t ) ~ ( a , b , t , < ) if a > t ( b , t ) B ( ~ , b , t , {z}K(z, [ ;b ~, t , ; j ) ,3 otherwise Proofi T h e idea, of course, is t o define K ( a , b , t , 3 by induction on the length of b . Let B*(a, b , t , c u ~ , ~ ) be the formula and let D(u ,4) be the formula I t is easy t o see t h a t

Bounded Arithmetic と計算量の根本問題

We consider foundational questions related to the de nition of functions by corecursion. This method is especially suited to functions into the greatest xed point of some monotone operator, and it is most applicable in the context of non-wellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [NWF] and the Corecursion Theorem of Barwise and Moss [VC]. We prove some results relating the two approaches, and we present a framework from which one can derive the results of both.

On the Foundations of Corecursion

This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR1, has its first-order programs characterize the poly-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR1-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook's and Leivant's implicit characterizations of poly-time. The time-restricting part is an affine version of Barber and Plotkin's DILL. Two semantics are constructed for ATR1. The first is a pruning of the naive denotational semantics for ATR1. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR1's time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR1 recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine.

http://www.cis.syr.edu/~royer/archive/ats-popl06.pdf

Adventures in time and space

Towards a model theory of Venn diagrams

Tor is one of the more popular systems for anonymizing near-real-time communications on the Internet. Borisov et al. [2007] proposed a denial-of-service-based attack on Tor (and related systems) that significantly increases the probability of compromising the anonymity provided. In this article, we analyze the effectiveness of the attack using both an analytic model and simulation. We also describe two algorithms for detecting such attacks, one deterministic and proved correct, the other probabilistic and verified in simulation.

Effectiveness and detection of denial-of-service attacks in tor

A logical system is studied whose well-formed representations consist of diagrams rather than formulas. The system, due to Shin [2, 3], is shown to be complete by an argument concerning maximally consistent sets of diagrams. The argument is complicated by the lack of a straight forward counterpart of atomic formulas for diagrams, and by the lack of a counterpart of negation for most diagrams.

Towards a model theory of diagrams

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