{"id":5325,"date":"2025-09-17T10:00:00","date_gmt":"2025-09-17T08:00:00","guid":{"rendered":"https:\/\/fragmentix.com\/?p=5325"},"modified":"2025-09-05T14:26:06","modified_gmt":"2025-09-05T12:26:06","slug":"secret-sharing-in-practice","status":"publish","type":"post","link":"https:\/\/fragmentix.com\/pt\/secret-sharing-in-practice\/","title":{"rendered":"Secret Sharing na pr\u00e1tica para uma prote\u00e7\u00e3o de dados dimension\u00e1vel, robusta e preparada para o futuro"},"content":{"rendered":"[et_pb_section admin_label=&#8221;section&#8221;]\n\t\t\t[et_pb_row admin_label=&#8221;row&#8221;]\n\t\t\t\t[et_pb_column type=&#8221;4_4&#8243;][et_pb_text admin_label=&#8221;Text&#8221;]\n<h5 class=\"wp-block-heading\"><strong>Secret Sharing<\/strong>:\u00a0<strong>The Technology Powering fragmentiX (Part 2)<\/strong><\/h5>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Beyond Shamir<\/h2>\n\n\n\n<p>In <a href=\"https:\/\/fragmentix.com\/what-is-secret-sharing\/\" data-type=\"post\" data-id=\"5287\">part one<\/a> of this blog post, we introduced the fundamentals of Shamir\u2019s Secret Sharing Scheme. While the scheme is powerful, real-world applications often demand more flexibility, efficiency, and resilience \u2013 especially in distributed cloud storage. In this second part, we\u2019ll take a closer look at some key extensions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Computationally Secure Secret Sharing (CSS): A Hybrid Approach for Efficiency<\/h2>\n\n\n\n<p>While Shamir\u2019s Secret Sharing offers strong security guarantees, it comes with a notable drawback: each share is as large as the original secret. This means that if you&#8217;re splitting a 1MB file into 10 shares, you&#8217;re looking at 10MB of total storage \u2013 an overhead that may become impractical or costly for large-scale systems.<\/p>\n\n\n\n<p>To address this, more space-efficient secret sharing schemes were introduced, with important contributions from researchers like Hugo Krawczyk <a id=\"_ftnref1\" href=\"#_ftn1\">[1]<\/a>. The core idea is to decouple the secret from its payload: instead of secret-sharing the entire data, one encrypts the data using a symmetric encryption algorithm (like <a href=\"https:\/\/de.wikipedia.org\/wiki\/Advanced_Encryption_Standard\" target=\"_blank\" rel=\"noopener\">AES<\/a>), and then applies Shamir\u2019s Secret Sharing only to the encryption key.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">How CSS Works<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Encrypt the secret using a randomly generated symmetric key.<\/li>\n\n\n\n<li>Apply Shamir\u2019s Secret Sharing to split the key into <em>n<\/em> shares with a threshold <em>k<\/em>.<\/li>\n\n\n\n<li>Distribute the encrypted secret using a space-efficient k-out-of-n distribution scheme (for example Rabin\u2019s Information Dispersal Algorithm <a id=\"_ftnref2\" href=\"#_ftn2\">[2]<\/a>) \u2013 similar in spirit to how <a href=\"https:\/\/en.wikipedia.org\/wiki\/RAID\" target=\"_blank\" rel=\"noopener\">RAID<\/a> systems distribute data across disks.<\/li>\n<\/ul>\n\n\n\n<p>This hybrid approach can dramatically reduces storage overhead. For example, if a 1\u202fMB file is distributed using a 7-out-of-10 scheme, the total storage required is only about 1.43\u202fMB &#8211; a significant improvement compared to the 10\u202fMB required by pure Shamir\u2019s Secret Sharing, where each of the 10 shares would be as large as the original file.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">CSS Security Considerations<\/h3>\n\n\n\n<p>Unlike pure Shamir\u2019s Secret Sharing, which offers information-theoretic security, this hybrid scheme\u2019s security depends on the strength of the symmetric cipher used. If the encryption algorithm is broken or the key is weak, the secret may be compromised. That said, modern symmetric ciphers like AES-256 are considered highly secure and also robust against quantum computers.<\/p>\n\n\n\n<p>Notably, even though encryption is used, no encryption keys need to be managed or stored. The key itself is secret-shared and reconstructed only when needed, eliminating the need for persistent key storage or key exchange protocols.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Key Benefits of CSS<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Storage efficiency: Only the key is secret-shared with Shamir\u2019s Secret Sharing, not the entire data.<\/li>\n\n\n\n<li>Scalability: Ideal for large files or systems with limited bandwidth.<\/li>\n\n\n\n<li>Simplified key management: The key is transient and distributed, not stored.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Robust Secret Sharing: A Resilient Approach for Real-World Cloud Security<\/h2>\n\n\n\n<p>In ideal conditions, secret sharing assumes that all participants are honest and all shares are intact. But real-world systems are rarely so cooperative. Shares can be corrupted due to hardware failures, transmission errors, or even deliberate tampering by adversaries. In such environments, Robust Secret Sharing becomes essential.<\/p>\n\n\n\n<p>Robust schemes are designed to tolerate faulty or malicious shares and still reconstruct the secret correctly. They extend the basic threshold model by adding mechanisms to detect and isolate invalid shares \u2013 ensuring that the reconstruction process is both accurate and secure.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Detecting Corrupted Shares with Hashes and MACs<\/h3>\n\n\n\n<p>One common approach to robustness is to attach cryptographic checksums to each share:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Hashes: A hash of the share can be computed and stored alongside it. During reconstruction, each share is re-hashed and compared to its stored hash. If the values don\u2019t match, the share is considered corrupted.<\/li>\n\n\n\n<li>Message Authentication Codes (MACs): MACs are more secure than plain hashes because they use a secret key to generate the checksum. This prevents adversaries from forging valid-looking shares. Each share is accompanied by its MAC, allowing participants to verify the integrity of the share before using it.<\/li>\n<\/ul>\n\n\n\n<p>The checksums of all shares are typically stored together with each individual share. This setup allows shares to verify each other, helping the system identify and discard invalid or malicious inputs.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Confidentiality Considerations in Robust Secret Sharing<\/h3>\n\n\n\n<p>While hashes and MACs improve integrity, they must be used with care &#8211; especially in schemes that aim for information-theoretic security. Publishing or attaching metadata to shares can inadvertently leak information that reduce entropy or hint at the secret\u2019s structure. If not properly designed, these additions could weaken the confidentiality guarantees of the underlying scheme.<\/p>\n\n\n\n<p>To preserve the strong security properties of Shamir\u2019s scheme, robustness mechanisms must be carefully integrated, ideally without introducing new assumptions or dependencies on computational hardness. A notable information-theoretically secure scheme was introduced by Rabin and Ben-Or <a id=\"_ftnref3\" href=\"#_ftn3\">[3]<\/a>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Real-World Use Case: Self-Healing Cloud Storage<\/h3>\n\n\n\n<p>In distributed cloud storage, robust secret sharing ensures that data can be recovered even if some storage nodes are compromised or malfunctioning. It\u2019s especially valuable in potentially hostile or unreliable networks where trust and reliability can\u2019t be guaranteed. fragmentiX systems implement robust secret sharing not only to detect corrupted shares, but also to automatically recover them \u2013 enabling self-healing capabilities that enhance resilience and reduce the need for manual intervention.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conclusion and Outlook: Future-Proofing Data with Secret Sharing<\/h2>\n\n\n\n<p>Secret sharing is more than a clever cryptographic trick \u2013 it\u2019s a foundational tool for building resilient, secure systems. From protecting encryption keys to enabling fault-tolerant cloud storage, its applications are vast and growing.<\/p>\n\n\n\n<p>At fragmentiX, we integrate these techniques into our distributed cloud storage solution to enhance both security and reliability. While the cryptographic foundations are decades old, their relevance is only growing, especially in a world where trust must be distributed and resilience is non-negotiable.<\/p>\n\n\n\n<p>As quantum computing and advanced cryptanalysis continue to challenge traditional encryption schemes, secret sharing stands out for its information-theoretic security and independence from computational hardness assumptions. This makes it a compelling choice for future-proofing sensitive data against both current and emerging threats.<\/p>\n\n\n\n<p>Whether you&#8217;re a business with sensitive data or just someone who values privacy, one thing\u2019s clear: sometimes, sharing is the safest way to keep a secret.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>References<\/strong><\/p>\n\n\n\n<p><a href=\"#_ftnref1\" id=\"_ftn1\">[1]<\/a>Krawczyk, H. (1994). <em>Secret Sharing Made Short<\/em>. In: Stinson, D.R. (eds) Advances in Cryptology \u2014 CRYPTO\u2019 93. CRYPTO 1993. Lecture Notes in Computer Science, vol 773. Springer, Berlin, Heidelberg. https:\/\/doi.org\/10.1007\/3-540-48329-2_12<\/p>\n\n\n\n<p><a href=\"#_ftnref2\" id=\"_ftn2\">[2]<\/a>Michael O. Rabin. 1989. <em>Efficient dispersal of information for security, load balancing, and fault tolerance<\/em>. J. ACM 36, 2 (April 1989), 335\u2013348. https:\/\/doi.org\/10.1145\/62044.62050<\/p>\n\n\n\n<p><a href=\"#_ftnref3\" id=\"_ftn3\">[3]<\/a> T. Rabin and M. Ben-Or. 1989. <em>Verifiable secret sharing and multiparty protocols with honest majority<\/em>. In Proceedings of the twenty-first annual ACM symposium on Theory of computing (STOC &#8217;89). Association for Computing Machinery, New York, NY, USA, 73\u201385. https:\/\/doi.org\/10.1145\/73007.73014<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Ready to protect your data from future threats?<\/h4>\n\n\n\n<p>\u27a1\ufe0f Get in touch for a&nbsp;<a href=\"https:\/\/fragmentix.com\/contact-us\/\">consultation or demo<\/a>&nbsp;tailored to your infrastructure.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Learn how fragmentiX works in detail:<\/h4>\n\n\n\n<p>\u27a1\ufe0f Read part 1 <a href=\"https:\/\/fragmentix.com\/what-is-secret-sharing\/\" data-type=\"post\" data-id=\"5287\">&#8220;What Is Secret Sharing? \u2013Shamir\u2019s Scheme Explained and Why It Matters for Data Security&#8221;<\/a><\/p>\n\n\n\n<p>\u27a1\ufe0f Explore our&nbsp;<a href=\"https:\/\/fragmentix.com\/products\/\">solutions<\/a>.<\/p>\n[\/et_pb_text][\/et_pb_column]\n\t\t\t[\/et_pb_row]\n\t\t[\/et_pb_section]","protected":false},"excerpt":{"rendered":"<p>Na primeira parte desta postagem do blog, apresentamos os fundamentos do esquema Secret Sharing de Shamir. Embora o esquema seja eficiente, os aplicativos do mundo real geralmente exigem mais flexibilidade, efici\u00eancia e resili\u00eancia, especialmente no armazenamento distribu\u00eddo em nuvem. Nesta segunda parte, daremos uma olhada mais de perto em algumas extens\u00f5es importantes.<\/p>","protected":false},"author":16,"featured_media":5296,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"<!-- wp:heading {\"level\":5} -->\n<h5 class=\"wp-block-heading\"><strong>Secret Sharing<\/strong>:\u00a0<strong>The Technology Powering fragmentiX (Part 2)<\/strong><\/h5>\n<!-- \/wp:heading -->\n\n<!-- wp:separator {\"className\":\"is-style-wide\"} -->\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<!-- \/wp:separator -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Beyond Shamir<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>In <a href=\"https:\/\/fragmentix.com\/what-is-secret-sharing\/\" data-type=\"post\" data-id=\"5287\">part one<\/a> of this blog post, we introduced the fundamentals of Shamir\u2019s Secret Sharing Scheme. While the scheme is powerful, real-world applications often demand more flexibility, efficiency, and resilience \u2013 especially in distributed cloud storage. In this second part, we\u2019ll take a closer look at some key extensions.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Computationally Secure Secret Sharing (CSS): A Hybrid Approach for Efficiency<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>While Shamir\u2019s Secret Sharing offers strong security guarantees, it comes with a notable drawback: each share is as large as the original secret. This means that if you're splitting a 1MB file into 10 shares, you're looking at 10MB of total storage \u2013 an overhead that may become impractical or costly for large-scale systems.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>To address this, more space-efficient secret sharing schemes were introduced, with important contributions from researchers like Hugo Krawczyk <a id=\"_ftnref1\" href=\"#_ftn1\">[1]<\/a>. The core idea is to decouple the secret from its payload: instead of secret-sharing the entire data, one encrypts the data using a symmetric encryption algorithm (like <a href=\"https:\/\/de.wikipedia.org\/wiki\/Advanced_Encryption_Standard\">AES<\/a>), and then applies Shamir\u2019s Secret Sharing only to the encryption key.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\">How CSS Works<\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:list -->\n<ul class=\"wp-block-list\"><!-- wp:list-item -->\n<li>Encrypt the secret using a randomly generated symmetric key.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Apply Shamir\u2019s Secret Sharing to split the key into <em>n<\/em> shares with a threshold <em>k<\/em>.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Distribute the encrypted secret using a space-efficient k-out-of-n distribution scheme (for example Rabin\u2019s Information Dispersal Algorithm <a id=\"_ftnref2\" href=\"#_ftn2\">[2]<\/a>) \u2013 similar in spirit to how <a href=\"https:\/\/en.wikipedia.org\/wiki\/RAID\">RAID<\/a> systems distribute data across disks.<\/li>\n<!-- \/wp:list-item --><\/ul>\n<!-- \/wp:list -->\n\n<!-- wp:paragraph -->\n<p>This hybrid approach can dramatically reduces storage overhead. For example, if a 1\u202fMB file is distributed using a 7-out-of-10 scheme, the total storage required is only about 1.43\u202fMB - a significant improvement compared to the 10\u202fMB required by pure Shamir\u2019s Secret Sharing, where each of the 10 shares would be as large as the original file.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\">CSS Security Considerations<\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>Unlike pure Shamir\u2019s Secret Sharing, which offers information-theoretic security, this hybrid scheme\u2019s security depends on the strength of the symmetric cipher used. If the encryption algorithm is broken or the key is weak, the secret may be compromised. That said, modern symmetric ciphers like AES-256 are considered highly secure and also robust against quantum computers.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Notably, even though encryption is used, no encryption keys need to be managed or stored. The key itself is secret-shared and reconstructed only when needed, eliminating the need for persistent key storage or key exchange protocols.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\">Key Benefits of CSS<\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:list -->\n<ul class=\"wp-block-list\"><!-- wp:list-item -->\n<li>Storage efficiency: Only the key is secret-shared with Shamir\u2019s Secret Sharing, not the entire data.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Scalability: Ideal for large files or systems with limited bandwidth.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Simplified key management: The key is transient and distributed, not stored.<\/li>\n<!-- \/wp:list-item --><\/ul>\n<!-- \/wp:list -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Robust Secret Sharing: A Resilient Approach for Real-World Cloud Security<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>In ideal conditions, secret sharing assumes that all participants are honest and all shares are intact. But real-world systems are rarely so cooperative. Shares can be corrupted due to hardware failures, transmission errors, or even deliberate tampering by adversaries. In such environments, Robust Secret Sharing becomes essential.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Robust schemes are designed to tolerate faulty or malicious shares and still reconstruct the secret correctly. They extend the basic threshold model by adding mechanisms to detect and isolate invalid shares \u2013 ensuring that the reconstruction process is both accurate and secure.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\">Detecting Corrupted Shares with Hashes and MACs<\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>One common approach to robustness is to attach cryptographic checksums to each share:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:list -->\n<ul class=\"wp-block-list\"><!-- wp:list-item -->\n<li>Hashes: A hash of the share can be computed and stored alongside it. During reconstruction, each share is re-hashed and compared to its stored hash. If the values don\u2019t match, the share is considered corrupted.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Message Authentication Codes (MACs): MACs are more secure than plain hashes because they use a secret key to generate the checksum. This prevents adversaries from forging valid-looking shares. Each share is accompanied by its MAC, allowing participants to verify the integrity of the share before using it.<\/li>\n<!-- \/wp:list-item --><\/ul>\n<!-- \/wp:list -->\n\n<!-- wp:paragraph -->\n<p>The checksums of all shares are typically stored together with each individual share. This setup allows shares to verify each other, helping the system identify and discard invalid or malicious inputs.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\">Confidentiality Considerations in Robust Secret Sharing<\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>While hashes and MACs improve integrity, they must be used with care - especially in schemes that aim for information-theoretic security. Publishing or attaching metadata to shares can inadvertently leak information that reduce entropy or hint at the secret\u2019s structure. If not properly designed, these additions could weaken the confidentiality guarantees of the underlying scheme.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>To preserve the strong security properties of Shamir\u2019s scheme, robustness mechanisms must be carefully integrated, ideally without introducing new assumptions or dependencies on computational hardness. A notable information-theoretically secure scheme was introduced by Rabin and Ben-Or <a id=\"_ftnref3\" href=\"#_ftn3\">[3]<\/a>.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\">Real-World Use Case: Self-Healing Cloud Storage<\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>In distributed cloud storage, robust secret sharing ensures that data can be recovered even if some storage nodes are compromised or malfunctioning. It\u2019s especially valuable in potentially hostile or unreliable networks where trust and reliability can\u2019t be guaranteed. fragmentiX systems implement robust secret sharing not only to detect corrupted shares, but also to automatically recover them \u2013 enabling self-healing capabilities that enhance resilience and reduce the need for manual intervention.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Conclusion and Outlook: Future-Proofing Data with Secret Sharing<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>Secret sharing is more than a clever cryptographic trick \u2013 it\u2019s a foundational tool for building resilient, secure systems. From protecting encryption keys to enabling fault-tolerant cloud storage, its applications are vast and growing.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>At fragmentiX, we integrate these techniques into our distributed cloud storage solution to enhance both security and reliability. While the cryptographic foundations are decades old, their relevance is only growing, especially in a world where trust must be distributed and resilience is non-negotiable.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>As quantum computing and advanced cryptanalysis continue to challenge traditional encryption schemes, secret sharing stands out for its information-theoretic security and independence from computational hardness assumptions. This makes it a compelling choice for future-proofing sensitive data against both current and emerging threats.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Whether you're a business with sensitive data or just someone who values privacy, one thing\u2019s clear: sometimes, sharing is the safest way to keep a secret.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><strong>References<\/strong><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><a href=\"#_ftnref1\" id=\"_ftn1\">[1]<\/a>Krawczyk, H. (1994). <em>Secret Sharing Made Short<\/em>. In: Stinson, D.R. (eds) Advances in Cryptology \u2014 CRYPTO\u2019 93. CRYPTO 1993. Lecture Notes in Computer Science, vol 773. Springer, Berlin, Heidelberg. https:\/\/doi.org\/10.1007\/3-540-48329-2_12<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><a href=\"#_ftnref2\" id=\"_ftn2\">[2]<\/a>Michael O. Rabin. 1989. <em>Efficient dispersal of information for security, load balancing, and fault tolerance<\/em>. J. ACM 36, 2 (April 1989), 335\u2013348. https:\/\/doi.org\/10.1145\/62044.62050<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><a href=\"#_ftnref3\" id=\"_ftn3\">[3]<\/a> T. Rabin and M. Ben-Or. 1989. <em>Verifiable secret sharing and multiparty protocols with honest majority<\/em>. In Proceedings of the twenty-first annual ACM symposium on Theory of computing (STOC '89). Association for Computing Machinery, New York, NY, USA, 73\u201385. https:\/\/doi.org\/10.1145\/73007.73014<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:separator {\"className\":\"is-style-wide\"} -->\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<!-- \/wp:separator -->\n\n<!-- wp:heading {\"level\":4} -->\n<h4 class=\"wp-block-heading\">Ready to protect your data from future threats?<\/h4>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>\u27a1\ufe0f Get in touch for a&nbsp;<a href=\"https:\/\/fragmentix.com\/contact-us\/\">consultation or demo<\/a>&nbsp;tailored to your infrastructure.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":4} -->\n<h4 class=\"wp-block-heading\">Learn how fragmentiX works in detail:<\/h4>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>\u27a1\ufe0f Read part 1 <a href=\"https:\/\/fragmentix.com\/what-is-secret-sharing\/\" data-type=\"post\" data-id=\"5287\">\"What Is Secret Sharing? \u2013Shamir\u2019s Scheme Explained and Why It Matters for Data Security\"<\/a><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\u27a1\ufe0f Explore our&nbsp;<a href=\"https:\/\/fragmentix.com\/products\/\">solutions<\/a>.<\/p>\n<!-- \/wp:paragraph -->","_et_gb_content_width":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[4],"tags":[69,74,102,60,104],"class_list":["post-5325","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-technology","tag-fragmentix","tag-its","tag-quantum-safe-storage","tag-secret-sharing","tag-secret-sharing-technology"],"_links":{"self":[{"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/posts\/5325","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/users\/16"}],"replies":[{"embeddable":true,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/comments?post=5325"}],"version-history":[{"count":4,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/posts\/5325\/revisions"}],"predecessor-version":[{"id":5340,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/posts\/5325\/revisions\/5340"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/media\/5296"}],"wp:attachment":[{"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/media?parent=5325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/categories?post=5325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fragmentix.com\/pt\/wp-json\/wp\/v2\/tags?post=5325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}