Moessner’s Theorem: An Exercise in Coinductive Reasoning in Coq

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Fast secure comparison for medium-sized integers and its application in binarized neural networks

In 1994, Feige, Kilian, and Naor proposed a simple protocol for secure 3-way comparison of integers a and b from the range [0, 2]. Their observation is that for (Formula Presented), the Legendre symbol (Formula Presented) coincides with the sign of x for (Formula Presented), thus reducing secure comparison to secure evaluation of the Legendre symbol. More recently, in 2011, Yu generalized this idea to handle secure comparisons for integers from substantially larger ranges [0, d], essentially by searching for primes for which the Legendre symbol coincides with the sign function on (Formula Presented). In this paper, we present new comparison protocols based on the Legendre symbol that additionally employ some form of error correction. We relax the prime search by requiring that the Legendre symbol encodes the sign function in a noisy fashion only. Practically, we use the majority vote over a window of (Formula Presented) adjacent Legendre symbols, for small positive integers k. Our technique significantly increases the comparison range: e.g., for a modulus of 60 bits, d increases by a factor of 2.8 (for (Formula Presented)) and 3.8 (for (Formula Presented)) respectively. We give a practical method to find primes with suitable noisy encodings. We demonstrate the practical relevance of our comparison protocol by applying it in a secure neural network classifier for the MNIST dataset. Concretely, we discuss a secure multiparty computation based on the binarized multi-layer perceptron of Hubara et al., using our comparison for the second and third layers

T-cell exhaustion and residency dynamics inform clinical outcomes in hepatocellular carcinoma

Background & aims: Despite recent translation of immunotherapies into clinical practice, the immunobiology of hepatocellular carcinoma (HCC), in particular the role and clinical relevance of exhausted and liver-resident T cells remain unclear. We therefore dissected the landscape of exhausted and resident T cell responses in the peripheral blood and tumor microenvironment of patients with HCC. Methods: Lymphocytes were isolated from the blood, tumor and tumor-surrounding liver tissue of patients with HCC (n = 40, n = 10 treated with anti-PD-1 therapy). Phenotype, function and response to anti-PD-1 were analyzed by mass and flow cytometry ex vivo and in vitro, tissue residence was further assessed by immunohistochemistry and imaging mass cytometry. Gene signatures were analyzed in silico. Results: We identified significant enrichment of heterogeneous populations of exhausted CD8+ T cells (TEX) in the tumor microenvironment. Strong enrichment of severely exhausted CD8 T cells expressing multiple immune checkpoints in addition to PD-1 was linked to poor progression-free and overall survival. In contrast, PD-1 was also expressed on a subset of more functional and metabolically active CD103+ tissue-resident memory T cells (TRM) that expressed few additional immune checkpoints and were associated with better survival. TEX enrichment was independent of BCLC stage, alpha-fetoprotein levels or age as a variable for progression-free survival in our cohort. These findings were in line with in silico gene signature analysis of HCC tumor transcriptomes from The Cancer Genome Atlas. A higher baseline TRM/TEX ratio was associated with disease control in anti-PD-1-treated patients. Conclusion: Our data provide information on the role of peripheral and intratumoral TEX-TRM dynamics in determining outcomes in patients with HCC. The dynamics between exhausted and liver-resident T cells have implications for immune-based diagnostics, rational patient selection and monitoring during HCC immunotherapies. Lay summary: The role of the immune response in hepatocellular carcinoma (HCC) remains unclear. T cells can mediate protection against tumor cells but are frequently dysfunctional and exhausted in cancer. We found that patients with a predominance of exhausted CD8+ T cells (TEX) had poor survival compared to patients with a predominance of tissue-resident memory T cells (TRM). This correlated with the molecular profile, metabolic and functional status of these cell populations. The enrichment of TEX was independently associated with prognosis in addition to disease stage, age and tumor markers. A high TRM proportion was also associated with better outcomes following checkpoint therapy. Thus, these T-cell populations are novel biomarkers with relevance in HCC