DoubleMod and SingleMod: Simple Randomized Secret-Key Encryption with Bounded Homomorphicity

An encryption relation f Z Z with decryption function f 1 is “group-homomorphic” if, for any suitable plaintexts x1 and x2, x1+x2 = f 1( f (x1)+f (x2)). It is “ring-homomorphic” if furthermore x1x2 = f 1( f (x1) f (x2)); it is “field-homomorphic” if furthermore 1=x1 = f 1( f (1=x1)). Such relations would support oblivious processing of encrypted data. We propose a simple randomized encryption relation f over the integers, called DoubleMod, which is “bounded ring-homomorphic” or what some call ”somewhat homomorphic.” Here, “bounded” means that the number of additions and multiplications that can be performed, while not allowing the encrypted values to go out of range, is limited (any pre-specified bound on the operation-count can be accommodated). Let R be any large integer. For any plaintext x 2 ZR, DoubleMod encrypts x as f (x) = x + au + bv, where a and b are randomly chosen integers in some appropriate interval, while (u; v) is the secret key. Here u > R2 is a large prime and the smallest prime factor of v exceeds u. With knowledge of the key, but not of a and b, the receiver decrypts the ciphertext by computing f 1(y) = (y mod v) mod u. DoubleMod generalizes an independent idea of van Dijk et al. 2010. We present and refine a new CCA1 chosen-ciphertext attack that finds the secret key of both systems (ours and van Dijk et al.’s) in linear time in the bit length of the security parameter. Under a known-plaintext attack, breaking DoubleMod is at most as hard as solving the Approximate GCD (AGCD) problem. The complexity of AGCD is not known. We also introduce the SingleMod field-homomorphic cryptosystems. The simplest SingleMod system based on the integers can be broken trivially. We had hoped, that if SingleMod is implemented inside non-Euclidean quadratic or higher-order fields with large discriminants, where GCD computations appear di cult, it may be feasible to achieve a desired level of security. We show, however, that a variation of our chosen-ciphertext attack works against SingleMod even in non-Euclidean fields

Promoting Historical Thinking in Schools through Low Fidelity, Low-Cost, Easily Reproduceable, Tangible and Embodied Interactions

Many ubiquitous technologies, such as augmented and virtual reality, virtual field trips, tangible environments and interactive installations are continuously being examined for learning about history. All these approaches strive to offer an inter-active, explorative, authentic, and exact regeneration of the past and, they are demanding both in regards to their cost and to their development requirements; hence, they are far from accessible in school contexts. In this study, we will ex-amine, whether interactive low-fidelity low-cost approximations of the real objects, and conditions of the past, together with whole-body interactions may enable historical understanding in a playful manner. To evaluate this claim, sixty-six (66) students of 6th grade played with an interactive environment with the characteristics mentioned, in twenty (20) groups. Afterwards, students were asked to complete a questionnaire assessing their historical understanding and their experience. Additionally, all groups participated in brief group interviews. Students were very positive towards the environment and considered it as enjoyable, intriguing and effective. They claimed that it is preferable to both traditional ways of classroom learning and learning devices like tablets and mobiles. According to our results, the requirements for seamless authenticity may not apply for primary school students and this conclusion opens up a lot of opportunities for moving away from traditional means of learning about history

Promoting Historical Thinking in Schools through Low Fidelity, Low-Cost, Easily Reproduceable, Tangible and Embodied Interactions

Many ubiquitous technologies, such as augmented and virtual reality, virtual field trips, tangible environments and interactive installations are continuously being examined for learning about history. All these approaches strive to offer an inter-active, explorative, authentic, and exact regeneration of the past and, they are demanding both in regards to their cost and to their development requirements; hence, they are far from accessible in school contexts. In this study, we will ex-amine, whether interactive low-fidelity low-cost approximations of the real objects, and conditions of the past, together with whole-body interactions may enable historical understanding in a playful manner. To evaluate this claim, sixty-six (66) students of 6th grade played with an interactive environment with the characteristics mentioned, in twenty (20) groups. Afterwards, students were asked to complete a questionnaire assessing their historical understanding and their experience. Additionally, all groups participated in brief group interviews. Students were very positive towards the environment and considered it as enjoyable, intriguing and effective. They claimed that it is preferable to both traditional ways of classroom learning and learning devices like tablets and mobiles. According to our results, the requirements for seamless authenticity may not apply for primary school students and this conclusion opens up a lot of opportunities for moving away from traditional means of learning about history