# CAPTCHA: Using Hard AI Problems For Security

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4 but was only meant to defeat off-the-shelf Optical Character Recognition (OCR) technology. (Coates et al [5], inspired by our work, and Xu et al [14] developed similar systems and provided more concrete analyses.) In 2000 [1], we introduced the notion of a captcha as well as several practical proposals for Automated Turing Tests. This paper is the first to conduct a rigorous investigation of Automated Turing Tests and to address the issue of proving that it is difficult to write a computer program that can pass the tests. This, in turn, leads to a discussion of using AI problems for security purposes, which has never appeared in the literature. We also introduce the first Automated Turing Tests not based on the difficulty of Optical Character Recognition. A related general interest paper [2] has been accepted by Communications of the ACM. That paper reports on our work, without formalizing the notions or providing security guarantees. 2 Definitions and Notation Let C be a probability distribution. We use [C] to denote the support of C. If P ( ) is a probabilistic program, we will denote by P r ( ) the deterministic program that results when P uses random coins r. Let (P, V ) be a pair of probabilistic interacting programs. We denote the output of V after the interaction between P and V with random coins u 1 and u 2, assuming this interaction terminates, by P u1, V u2 (the subscripts are omitted in case the programs are deterministic). A program V is called a test if for all P and u 1, u 2, the interaction between P u1 and V u2 terminates and P u1, V u2 {accept, reject}. We call V the verifier or tester and any P which interacts with V the prover. Definition 1. Define the success of an entity A over a test V by Succ V A = Pr r,r [ A r, V r = accept]. We assume that A can have precise knowledge of how V works; the only piece of information that A can t know is r, the internal randomness of V. CAPTCHA Intuitively, a captcha is a test V over which most humans have success close to 1, and for which it is hard to write a computer program that has high success over V. We will say that it is hard to write a computer program that has high success over V if any program that has high success over V can be used to solve a hard AI problem. Definition 2. A test V is said to be (α, β)-human executable if at least an α portion of the human population has success greater than β over V.

5 Notice that a statement of the form V is (α, β)-human executable can only be proven empirically. Also, the success of different groups of humans might depend on their origin language or sensory disabilities: color-blind individuals, for instance, might have low success on tests that require the differentiation of colors. Definition 3. An AI problem is a triple P = (S, D, f), where S is a set of problem instances, D is a probability distribution over the problem set S, and f : S {0, 1} answers the instances. Let δ (0, 1]. We require that for an α > 0 fraction of the humans H, Pr x D [H(x) = f(x)] > δ. Definition 4. An AI problem P is said to be (δ, τ)-solved if there exists a program A, running in time at most τ on any input from S, such that Pr [A r(x) = f(x)] δ. x D,r (A is said to be a (δ, τ) solution to P.) P is said to be a (δ, τ)-hard AI problem if no current program is a (δ, τ) solution to P, and the AI community agrees it is hard to find such a solution. Definition 5. A (α, β, η)-captcha is a test V that is (α, β)-human executable, and which has the following property: There exists a (δ, τ)-hard AI problem P and a program A, such that if B has success greater than η over V then A B is a (δ, τ) solution to P. (Here A B is defined to take into account B s running time too.) We stress that V should be a program whose code is publicly available. Remarks 1. The definition of an AI problem as a triple (S, D, f) should not be inspected with a philosophical eye. We are not trying to capture all the problems that fall under the umbrella of Artificial Intelligence. We want the definition to be easy to understand, we want some AI problems to be captured by it, and we want the AI community to agree that these are indeed hard AI problems. More complex definitions can be substituted for Definition 3 and the rest of the paper remains unaffected. 2. A crucial characteristic of an AI problem is that a certain fraction of the human population be able to solve it. Notice that we don t impose a limit on how long it would take humans to solve the problem. All that we require is that some humans be able to solve it (even if we have to assume they will live hundreds of years to do so). The case is not the same for captchas. Although our definition says nothing about how long it should take a human to solve a captcha, it is preferable for humans to be able to solve captchas in a very short time. captchas which take a long time for humans to solve are probably useless for all practical purposes.

8 3 Two AI Problem Families In this section we introduce two families of AI problems that can be used to construct captchas. The section can be viewed as a precise statement of the kind of hardness assumptions that our cryptographic protocols are based on. We stress that solutions to the problems will also be shown to be useful. For the purposes of this paper, we define an image as an h w matrix (h for height and w for width) whose entries are pixels. A pixel is defined as a triple of integers (R, G, B), where 0 R, G, B M for some constant M. An image transformation is a function that takes as input an image and outputs another image (not necessarily of the same width and height). Examples of image transformations include: turning an image into its black-and-white version, changing its size, etc. Let I be a distribution on images and T be a distribution on image transformations. We assume for simplicity that if i, i [I] and i i then T (i) T (i ) for any T, T [T ]. (Recall that [I] denotes the support of I.) Problem Family (P1). Consider the following experiment: choose an image i I and choose a transformation t T ; output t(i). P1 I,T consists of writing a program that takes t(i) as input and outputs i (we assume that the program has precise knowledge of T and I). More formally, let S I,T = {t(i) : t [T ] and i [I]}, D I,T be the distribution on S I,T that results from performing the above experiment and f I,T : S I,T [I] be such that f I,T (t(i)) = i. Then P1 I,T = (S I,T, D I,T, f I,T ). Problem Family (P2). In addition to the distributions I and T, let L be a finite set of labels. Let λ : [I] L compute the label of an image. The set of problem instances is S I,T = {t(i) : t [T ] and i [I]}, and the distribution on instances D I,T is the one induced by choosing i I and t T. Define g I,T,λ so that g I,T,λ (t(i)) = λ(i). Then P2 I,T,λ = (S I,T, D I,T, g I,T,λ ) consists of writing a program that takes t(i) as input and outputs λ(i). Remarks 1. Note that a (δ, τ) solution A to an instance of P1 also yields a (δ, τ + τ ) solution to an instance of P2 (where δ δ and τ log [I] is the time that it takes to compute λ), specifically, computing λ(a(x)). However, this may be unsatisfactory for small δ, and we might hope to do better by restricting to a smaller set of labels. Conversely, P1 can be seen as a special case of P2 with λ the identity function and L = [I]. Formally, problem families P1 and P2 can be shown to be isomorphic. Nonetheless, it is useful to make a distinction here because in some applications it appears unnatural to talk about labels. 2. We stress that in all the instantiations of P1 and P2 that we consider, I, T and, in the case of P2, λ, will be such that humans have no problem solving P1 I,T and P2 I,T,λ. That is, [T ] is a set of transformations that humans

11 come from I, guess that t(j) is a transformation of i; otherwise guess that it isn t. The reason for the asymmetry is to make the proof of Lemma 1 easier to follow. We note that a stronger captcha can be built by choosing i from the distribution I restricted to the set [I] {j}. 3. The intuition for why matcha plays another round when i j and res = 0 is that we are trying to convert high success against matcha into high success in solving P1; a program that solves P1 by comparing t(j) to every image in [I] will encounter that most of the images in [I] are different from t(j). 4. In the following Lemma we assume that a program with high success over M always terminates with a response in time at most τ. Any program which does not satisfy this requirement can be rewritten into one which does, by stopping after τ time and sending the response 1, which never decreases the success probability. We also assume that the unit of time that matcha uses is the same as one computational step. Lemma 1. Any program that has success greater than η over M = (I, T, τ) can be used to (δ, τ [I] )-solve P1 I,T, where δ η [I] (1 η). Proof. Let B be a program that runs in time at most τ and has success σ B η over M. Using B we construct a program A B that is a (δ, τ [I] )-solution to P1 I,T. The input to A B will be an image, and the output will be another image. On input j, A B will loop over the entire database of images of M (i.e., the set [I]), each time feeding B the pair of images (i, j), where i [I]. Afterwards, A B collects all the images i [I] on which B returned 1 (i.e., all the images in [I] that B thinks j is a transformation of). Call the set of these images S. If S is empty, then A B returns an element chosen uniformly from [I]. Otherwise, it picks an element l of S uniformly at random. We show that A B is a (δ, τ [I] )-solution to P1. Let p 0 = Pr T,I,r [B r (i, t(i)) = 0] and let p 1 = Pr T,j I,i,r [B r (i, t(j)) = 1]. Note that Pr r,r [ M r, B r = reject] = 1 σ B = p p 1 + (1 p 1 )(1 σ B ) = p 0 + p 1, p 1 which gives σ B 1 p 0 and p 1 2(1 σ B ). Hence:

12 Pr T,I,r [AB r (t(j)) = j] = n Pr T,I,r [AB r (t(j)) = j S = s] Pr [ S = s] (1) T,I s=1 n s=1 1 p 0 s 1 p 0 E T,I [ S ] Pr [ S = s] (2) T,I (3) 1 p [I] p 1 (4) σ B [I] (1 σ B ) (5) (2) follows by the definition of the procedure A B, (3) follows by Jensen s inequality and the fact that f(x) = 1/x is concave, (4) follows because E T,I [ S ] 1 + Pr(B(i, t(j)) = 1) I,r i j and (5) follows by the inequalities for p 0, p 1 and σ B given above. This completes the proof. Theorem 1. If P1 I,T is (δ, τ [I] )-hard and M = (I, T, τ) is (α, β)-human executable, then M is a (α, β, (2 [I] )δ 2δ [I] )-captcha. 4.2 PIX An instance P2 I,T,λ can sometimes be used almost directly as a captcha. For instance, if I is a distribution over images containing a single word and λ maps an image to the word contained in it, then P2 I,T,λ can be used directly as a captcha. Similarly, if all the images in [I] are pictures of simple concrete objects and λ maps an image to the object that is contained in the image, then P2 I,T,λ can be used as a captcha. Formally, a pix instance is a tuple X = (I, T, L, λ, τ). The pix verifier works as follows. First, V draws i I, and t T. V then sends to P the message (t(i), L), and sets a timer for τ. P responds with a label l L. V accepts if l = λ(i) and its timer has not expired, and rejects otherwise. Theorem 2. If P2 I,T,λ is (δ, τ)-hard and X = (I, T, L, λ, τ) is (α, β)-human executable, then X is a (α, β, δ)-captcha. Various instantiations of pix are in use at major internet portals, like Yahoo! and Hotmail. Other less conventional ones, like Animal-PIX, can be found at Animal-PIX presents the prover with a distorted picture of a common animal (like the one shown below) and asks it to choose between twenty different possibilities (monkey, horse, cow, et cetera).

13 Fig. 2. Animal-Pix. 5 An Application: Robust Image-Based Steganography We detail a useful application of (δ, τ)-solutions to instantiations of P1 and P2 (other than reading slightly distorted text, which was mentioned before). We hope to convey by this application that our problems were not chosen just because they can create captchas but because they in fact have applications related to security. Our problems also serve to illustrate that there is a need for better AI in security as well. Areas such a Digital Rights Management, for instance, could benefit from better AI: a program that can find slightly distorted versions of original songs or images on the world wide web would be a very useful tool for copyright owners. There are many applications of solutions to P1 and P2 that we don t mention here. P1, for instance, is interesting in its own right and a solution for the instantiation when I is a distribution on images of works of art would benefit museum curators, who often have to answer questions such as what painting is this a photograph of? Robust Image-Based Steganography. Robust Steganography is concerned with the problem of covertly communicating messages on a public channel which is subject to modification by a restricted adversary. For example, Alice may have some distribution on images which she is allowed to draw from and send to Bob; she may wish to communicate additional information with these pictures, in such a way that anyone observing her communications can not detect this additional information. The situation may be complicated by an adversary who transforms all transmitted images in an effort to remove any hidden information. In this section we will show how to use (δ, τ)-solutions to instantiations of P1 I,T or P2 I,T,λ to implement a secure robust steganographic protocol for image channels with distribution I, when the adversary chooses transformations from T. Note that if we require security for arbitrary I, T, we will require a (δ, τ)-solution to P1 for arbitrary I, T ; if no solution works for arbitrary (I, T ) this implies the existence of specific I, T for which P1 is still hard. Thus either our stegosystem can be implemented by

14 computers for arbitrary image channels or their is a (non-constructive) hard AI problem that can be used to construct a captcha. The results of this subsection can be seen as providing an implementation of the supraliminal channel postulated by Craver [6]. Indeed, Craver s observation that the adversary s transformations should be restricted to those which do not significantly impact human interpretation of the images (because the adversary should not unduly burden innocent correspondents) is what leads to the applicability of our hard AI problems. Steganography Definitions Fix a distribution over images I, and a set of keys K. A steganographic protocol or stegosystem for I is a pair of efficient probabilistic algorithms (SE, SD) where SE : K {0, 1} [I] l and SD : K [I] l {0, 1}, which have the additional property that Pr K,r,r [SD r (K, SE r (K, σ)) σ] is negligible (in l and K ) for any σ {0, 1}. We will describe a protocol for transmitting a single bit σ {0, 1}, but it is straightforward to extend our protocol and proofs by serial composition to any message in {0, 1} with at most linear decrease in security. Definition 6. A stegosystem is steganographically secret for I if the distributions {SE r (K, σ) : K K, r {0, 1} } and I l are computationally indistinguishable for any σ {0, 1}. Steganographic secrecy ensures that an eavesdropper cannot distinguish traffic produced by SE from I. Alice, however, is worried about a somewhat malicious adversary who transforms the images she transmits to Bob. This adversary is restricted by the fact that he must transform the images transmitted between many pairs of correspondents, and may not transform them in ways so that they are unrecognizable to humans, since he may not disrupt the communications of legitimate correspondents. Thus the adversary s actions, on seeing the image i, are restricted to selecting some transformation t according to a distribution T, and replacing i by t(i). Denote by t 1...l T l the action of independently selecting l transformations according to T, and denote by t 1...l (i 1...l ) the action of element-wise applying l transformations to l images. Definition 7. A stegosystem (SE, SD) is steganographically robust against T if it is steganographically secret and Pr [SD r (K, t 1...l (SE r (K, σ))) σ] t 1...l T l,r,r,k is negligible (in K and l) for any σ {0, 1}. Let F : K {1,..., l} L {0, 1} be a pseudorandom function family. We assume that λ : [I] L is efficiently computable and A : S P2 L is a (δ, τ)-solution to P2 I,T,λ as defined above (recall that P1 is a special case of P2), i.e. A operates in time τ and Pr [A r(t(i)) = λ(i)] δ. t,i,r

15 Let c = Pr i,j I [λ(i) = λ(j)]. We require that c < 1 (that is, we require that there is enough variability in the labels of the images to be useful for communication). Notice that L can simply be equal to [I] and λ can be the identity function (in case the images in [I] have no labels as in P1). We prove in the Appendix that the following construction is an efficient, robust stegosystem for T. Construction 1 Procedure SE: Input: K K, σ {0, 1} for j = 1... l do draw d 0 I, d 1 I if F K (j, λ(d 0 )) = σ then set i j = d 0 else set i j = d 1 Output: i 1, i 2,..., i l Procedure SD: Input: K K, i 1...l [I]l for j = 1... l do set σ j = F K (j, A(i j )) Output: majority(σ 1,..., σ l ) Proposition 1. Construction 1 is steganographically secret and robust for I, T. The proof of Proposition 1 is similar in flavor to those of Hopper, Langford and von Ahn [7] and relies on the fact that when A returns the correct solution on received image i j, the recovered bit σ j is equal to the intended bit σ with probability approximately (1 c) and otherwise σ j = σ with probability 1/2; therefore the probability that the majority of the σ j are incorrect is negligible in l. For details, see the Appendix. Remarks 1. Better solutions to P2 I,T,λ imply more efficient stegosystems: if δ is larger, then l can be smaller and less images need to be transmitted to send a bit secretively and robustly. 2. Since we assume that P2 I,T,λ (or, as it might be the case, P1 I,T ) is easy for humans, our protocol could be implemented as a cooperative effort between the human recipient and the decoding procedure (without the need for a solution to P1 I,T or P2 I,T,λ ). However, decoding each bit of the secret message will require classifying many images, so that a human would likely fail to complete the decoding well before any sizeable hidden message could be extracted (this is especially true in case we are dealing with P1 I,T and a large set [I]: a human would have to search the entire set [I] as many as l times for each transmitted bit). Thus to be practical, a (δ, τ)-solution (for small τ) to P1 I,T or P2 I,T,λ will be required. 6 Discussion and Closing Remarks Interaction with the AI community A primary goal of the captcha project is to serve as a challenge to the Artificial Intelligence community. We believe that having a well-specified set of goals will

17 References 1. Luis von Ahn, Manuel Blum, Nicholas J. Hopper and John Langford. The CAPTCHA Web Page: Luis von Ahn, Manuel Blum and John Langford. Telling Humans and Computers Apart (Automatically) or How Lazy Cryptographers do AI. To appear in Communications of the ACM. 3. Mihir Bellare, Russell Impagliazzo and Moni Naor. Does Parallel Repetition Lower the Error in Computationally Sound Protocols? In 38th IEEE Symposium on Foundations of Computer Science (FOCS 97), pages IEEE Computer Society, Mikhail M. Bongard. Pattern Recognition. Spartan Books, Rochelle Park NJ, A. L. Coates, H. S. Baird, and R. J. Fateman. Pessimal Print: A Reverse Turing Test. In Proceedings of the International Conference on Document Analysis and Recognition (ICDAR 01), pages Seattle WA, Scott Craver. On Public-key Steganography in the Presence of an Active Warden. In Proceedings of the Second International Information Hiding Workshop, pages Springer, Nicholas J. Hopper, John Langford and Luis von Ahn. Provably Secure Steganography. In Advances in Cryptology, CRYPTO 02, volume 2442 of Lecture Notes in Computer Science, pages Santa Barbara, CA, M. D. Lillibridge, M. Abadi, K. Bharat, and A. Broder. Method for selectively restricting access to computer systems. US Patent 6,195,698. Applied April 1998 and Approved February Greg Mori and Jitendra Malik. Breaking a Visual CAPTCHA. Unpublished Manuscript, Available electronically: Moni Naor. Verification of a human in the loop or Identification via the Turing Test. Unpublished Manuscript, Available electronically: Benny Pinkas and Tomas Sander. Securing Passwords Against Dictionary Attacks. In Proceedings of the ACM Computer and Security Conference (CCS 02), pages ACM Press, November S. Rice, G. Nagy, and T. Nartker. Optical Character Recognition: An Illustrated Guide to the Frontier. Kluwer Academic Publishers, Boston, Adi Shamir and Eran Tromer. Factoring Large Numbers with the TWIRL Device. Unpublished Manuscript, Available electronically: J. Xu, R. Lipton and I. Essa. Hello, are you human. Technical Report GIT-CC , Georgia Institute of Technology, November A Proof of Proposition 1 Lemma 1. Construction 1 is steganographically secret. Proof. Consider for any 1 j l and x [I] the probability ρ j x that i j = x, i.e. ρ j x = Pr[i j = x]. The image x is returned in the jth step only under one of the following conditions:

18 1. D 0 : d 0 = x and F K (j, λ(d 0 )) = σ ; or 2. D 1 : d 1 = x and F K (j, λ(d 0 )) = 1 σ Note that these events are mutually exclusive, so that ρ j x = Pr[D 0] + Pr[D 1 ]. Suppose that we replace F K by a random function f : {1,..., l} L {0, 1}. Then we have that Pr f,d0 [D 0 ] = 1 2 Pr I[x] by independence of f and d 0, and Pr f,d1 [D 1 ] = 1 2 Pr I[x], by the same reasoning. Thus ρ j x = Pr I [x] when F K is replaced by a random function. Further, for a random function the ρ j x are all independent. Thus for any σ {0, 1} we see that SE(K, σ) and I l are computationally indistinguishable, by the pseudorandomness of F. Lemma 2. Construction 1 is steganographically robust for T. Proof. Suppose we prove a constant bound ρ > 1 2 such that for all j, Pr[σ j = σ] > ρ. Then by a Chernoff bound we will have that Pr[majority(σ 1,..., σ l ) σ] is negligible, proving the lemma. Consider replacing F K for a random K with a randomly chosen function f : {1,..., l} L {0, 1} in SE, SD. We would like to assess the probability ρ j = Pr[σ j = σ]. Let A j be the event A(i j ) = λ(i j) and A j be the event A(i j ) λ(i j), and write λ(d j b ) as lj b. We have the following: Pr f,i j,t [σ j = σ] = Pr[σ j = σ A j ] Pr[A j ] + Pr[σ j = σ A j ] Pr[A j ] (6) = δ Pr[σ j = σ A j ] + 1 (1 δ) (7) 2 = δ(pr[f(j, l j 0 ) = σ or (f(j, lj 0 ) = 1 σ and f(j, 1 δ lj 1 )) = σ)] + 2 (8) = δ( Pr[f(j, lj 0 ) = 1 σ and f(j, lj 1 ) = σ and lj 0 lj 1 ]) + 1 δ 2 (9) = δ( δ (1 c)) (10) = δ (1 c) (11) 4 Here (7) follows because if A(i j ) = l λ(i j) then Pr f [f(j, l) = σ] = 1 2, and (8) follows because if A(i j ) = λ(i j), then f(j, A(i j )) = σ iff f(j, λ(i j)) = 0; the expression results from expanding the event f(j, λ(i j )) = 0 with the definition of i j in the encoding routine. For any constant δ > 0, a Chernoff bound implies that the probability of decoding failure is negligible in l when F K is a random function. Thus the pseudorandomness of F K implies that the probability of decoding failure for Construction 1 is also negligible, proving the lemma.

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