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	<title>cryptanalysis &#8211; Noise</title>
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	<link>https://noise.getoto.net</link>
	<description>The collective thoughts of the interwebz</description>
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		<title>New Cryptanalysis of the Fiat-Shamir Protocol</title>
		<link>https://noise.getoto.net/2025/09/09/new-cryptanalysis-of-the-fiat-shamir-protocol/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Tue, 09 Sep 2025 11:02:00 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[hashes]]></category>
		<category><![CDATA[protocols]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=70685</guid>

					<description><![CDATA[<p>A couple of months ago, a <a href="https://eprint.iacr.org/2025/118">new paper</a> demonstrated some new attacks against the Fiat-Shamir transformation. <i>Quanta</i> published a <a href="https://www.quantamagazine.org/computer-scientists-figure-out-how-to-prove-lies-20250709/">good article</a> that explains the results.</p>
<p>This is a pretty exciting paper from a theoretical perspective, but I don’t see it leading to any practical real-world cryptanalysis. The fact that there are some weird circumstances that result in Fiat-Shamir insecurities isn’t new—many dozens of papers have been published about it since 1986. What this new result does is extend this known problem to slightly less weird (but still highly contrived) situations. But it’s a completely different matter to extend these sorts of attacks to “natural” situations...</p>]]></description>
		
		
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		<title>1965 Cryptanalysis Training Workbook Released by the NSA</title>
		<link>https://noise.getoto.net/2025/09/02/1965-cryptanalysis-training-workbook-released-by-the-nsa/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Tue, 02 Sep 2025 11:08:48 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[history of cryptography]]></category>
		<category><![CDATA[NSA]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=70644</guid>

					<description><![CDATA[<p>In the early 1960s, National Security Agency cryptanalyst and cryptanalysis instructor Lambros D. Callimahos coined the term “Stethoscope” to describe a diagnostic computer program used to unravel the internal structure of pre-computer ciphertexts. The term appears in the newly declassified September 1965 document <i><a href="https://www.governmentattic.org/59docs/NSAlDCCDAC1965.pdf">Cryptanalytic Diagnosis with the Aid of a Computer</a></i>, which compiled 147 listings from this tool for Callimahos’s <a href="https://ia601207.us.archive.org/22/items/Legacy_Callimahos-nsa/Legacy_Callimahos.pdf">course</a>, <a href="https://www.nsa.gov/portals/75/documents/news-features/declassified-documents/cryptologic-spectrum/Callimahos_Course.pdf">CA-400: NSA Intensive Study Program in General Cryptanalysis</a>.</p>
<p>The listings in the report are printouts from the Stethoscope program, run on the NSA’s Bogart computer, showing statistical and structural data extracted from encrypted messages, but the encrypted messages themselves are not included. They were used in NSA training programs to teach analysts how to interpret ciphertext behavior without seeing the original message...</p>]]></description>
		
		
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		<title>The NSA’s “Fifty Years of Mathematical Cryptanalysis (1937–1987)”</title>
		<link>https://noise.getoto.net/2025/05/19/the-nsas-fifty-years-of-mathematical-cryptanalysis-1937-1987/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Mon, 19 May 2025 11:06:23 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[mathematics]]></category>
		<category><![CDATA[NSA]]></category>
		<category><![CDATA[reports]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=70269</guid>

					<description><![CDATA[In response to a FOIA request, the NSA released &#8220;Fifty Years of Mathematical Cryptanalysis (1937-1987),&#8221; by Glenn F. Stahly, with a lot of redactions.
Weirdly, this is the second time the NSA has declassified the document. John Young got a ...]]></description>
		
		
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		<title>Improvements in Brute Force Attacks</title>
		<link>https://noise.getoto.net/2025/03/17/improvements-in-brute-force-attacks/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Mon, 17 Mar 2025 15:09:57 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[keys]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=70019</guid>

					<description><![CDATA[<p>New paper: “<a href="https://tosc.iacr.org/index.php/ToSC/article/view/12078/11919">GPU Assisted Brute Force Cryptanalysis of GPRS, GSM, RFID, and TETRA: Brute Force Cryptanalysis of KASUMI, SPECK, and TEA3</a>.”</p>
<blockquote><p><b>Abstract:</b> Key lengths in symmetric cryptography are determined with respect to the brute force attacks with current technology. While nowadays at least 128-bit keys are recommended, there are many standards and real-world applications that use shorter keys. In order to estimate the actual threat imposed by using those short keys, precise estimates for attacks are crucial.</p>
<p>In this work we provide optimized implementations of several widely used algorithms on GPUs, leading to interesting insights on the cost of brute force attacks on several real-word applications...</p></blockquote>]]></description>
		
		
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		<title>Implementing Cryptography in AI Systems</title>
		<link>https://noise.getoto.net/2025/02/21/implementing-cryptography-in-ai-systems/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Fri, 21 Feb 2025 15:33:49 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[AI]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69951</guid>

					<description><![CDATA[<p>Interesting research: “<a href="https://eprint.iacr.org/2025/288">How to Securely Implement Cryptography in Deep Neural Networks</a>.”</p>
<blockquote><p><b>Abstract:</b> The wide adoption of deep neural networks (DNNs) raises the question of how can we equip them with a desired cryptographic functionality (e.g, to decrypt an encrypted input, to verify that this input is authorized, or to hide a secure watermark in the output). The problem is that cryptographic primitives are typically designed to run on digital computers that use Boolean gates to map sequences of bits to sequences of bits, whereas DNNs are a special type of analog computer that uses linear mappings and ReLUs to map vectors of real numbers to vectors of real numbers. This discrepancy between the discrete and continuous computational models raises the question of what is the best way to implement standard cryptographic primitives as DNNs, and whether DNN implementations of secure cryptosystems remain secure in the new setting, in which an attacker can ask the DNN to process a message whose “bits” are arbitrary real numbers...</p></blockquote>]]></description>
		
		
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		<title>New Advances in the Understanding of Prime Numbers</title>
		<link>https://noise.getoto.net/2024/12/18/new-advances-in-the-understanding-of-prime-numbers/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Wed, 18 Dec 2024 16:40:21 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[mathematics]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69704</guid>

					<description><![CDATA[Really interesting research into the structure of prime numbers. Not immediately related to the cryptanalysis of prime-number-based public-key algorithms, but every little bit matters.
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		<title>Matthew Green on Telegram’s Encryption</title>
		<link>https://noise.getoto.net/2024/08/28/matthew-green-on-telegrams-encryption/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Wed, 28 Aug 2024 11:00:40 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[encryption]]></category>
		<category><![CDATA[Telegram]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69302</guid>

					<description><![CDATA[Matthew Green wrote a really good blog post on what Telegram&#8217;s encryption is and is not.
EDITED TO ADD (8/28): Another good explainer from Kaspersky.
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		<title>On the Voynich Manuscript</title>
		<link>https://noise.getoto.net/2024/08/13/on-the-voynich-manuscript/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Tue, 13 Aug 2024 11:04:21 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[history of cryptography]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69245</guid>

					<description><![CDATA[<p>Really interesting <a href="https://www.theatlantic.com/magazine/archive/2024/09/decoding-voynich-manuscript/679157/?gift=YFkW3a8mqv4T0YBMneIYIuIiYZJAqQJorEylZzhFIOw&#38;utm_source=copy-link&#38;utm_medium=social&#38;utm_campaign=share&#38;fbclid=IwY2xjawEhtldleHRuA2FlbQIxMQABHdyEbPaL8wyhs9wMtkGXHfevH3pYDJ2kW9Oax8-NaxAEyKrmldht_ShcSg_aem_gPeUGAVQrTw8m61YZhwgig">article</a> on the ancient-manuscript scholars who are applying their techniques to the Voynich Manuscript.</p>
<p>No one has been able to understand the writing yet, but there are some new understandings:</p>
<blockquote><p>Davis presented her findings at the medieval-studies conference and <a href="https://muse.jhu.edu/pub/56/article/754633/pdf">published them in 2020</a> in the journal <i>Manuscript Studies</i>. She had hardly solved the Voynich, but she’d opened it to new kinds of investigation. If five scribes had come together to write it, the manuscript was probably the work of a community, rather than of a single deranged mind or con artist. Why the community used its own language, or code, remains a mystery. Whether it was a cloister of alchemists, or mad monks, or a group like the medieval Béguines—a secluded order of Christian women—required more study. But the marks of frequent use signaled that the manuscript served some routine, perhaps daily function...</p></blockquote>]]></description>
		
		
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		<title>Security Analysis of the EU’s Digital Wallet</title>
		<link>https://noise.getoto.net/2024/06/27/security-analysis-of-the-eus-digital-wallet/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Thu, 27 Jun 2024 11:06:32 +0000</pubDate>
				<category><![CDATA[credentials]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[eu]]></category>
		<category><![CDATA[identification]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69102</guid>

					<description><![CDATA[A group of cryptographers have analyzed the eiDAS 2.0 regulation (electronic identification and trust services) that defines the new EU Digital Identity Wallet.
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		<title>Breaking the M-209</title>
		<link>https://noise.getoto.net/2024/06/25/breaking-the-m-209/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Tue, 25 Jun 2024 11:02:32 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[history of cryptography]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69097</guid>

					<description><![CDATA[Interesting paper about a German cryptanalysis machine that helped break the US M-209 mechanical ciphering machine.
The paper contains a good description of how the M-209 works.
]]></description>
		
		
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		<title>Recovering Public Keys from Signatures</title>
		<link>https://noise.getoto.net/2024/06/20/recovering-public-keys-from-signatures/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Thu, 20 Jun 2024 11:10:53 +0000</pubDate>
				<category><![CDATA[anonymity]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[keys]]></category>
		<category><![CDATA[signatures]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69066</guid>

					<description><![CDATA[Interesting summary of various ways to derive the public key from digitally signed files.
Normally, with a signature scheme, you have the public key and want to know whether a given signature is valid. But what if we instead have a message and a signat...]]></description>
		
		
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		<title>Demo of AES GCM Misuse Problems</title>
		<link>https://noise.getoto.net/2024/06/14/demo-of-aes-gcm-misuse-problems/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Fri, 14 Jun 2024 11:05:35 +0000</pubDate>
				<category><![CDATA[aes]]></category>
		<category><![CDATA[algorithms]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[encryption]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=69025</guid>

					<description><![CDATA[This is  really neat demo of the security problems arising from reusing nonces with a symmetric cipher in GCM mode.
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		<title>Rare Interviews with Enigma Cryptanalyst Marian Rejewski</title>
		<link>https://noise.getoto.net/2024/05/03/rare-interviews-with-enigma-cryptanalyst-marian-rejewski/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Fri, 03 May 2024 11:10:08 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[enigma]]></category>
		<category><![CDATA[history of cryptography]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[war]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68858</guid>

					<description><![CDATA[The Polish Embassy has posted a series of short interview segments with Marian Rejewski, the first person to crack the Enigma.
Details from his biography.
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		<title>New Lattice Cryptanalytic Technique</title>
		<link>https://noise.getoto.net/2024/04/15/new-lattice-cryptanalytic-technique/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Mon, 15 Apr 2024 11:04:50 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[quantum cryptography]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68788</guid>

					<description><![CDATA[<p>A <a href="https://eprint.iacr.org/2024/555">new paper</a> presents a polynomial-time quantum algorithm for solving certain hard lattice problems. This could be a big deal for post-quantum cryptographic algorithms, since many of them base their security on hard lattice problems.</p>
<p>A few things to note. One, this paper has not yet been peer reviewed. As <a href="https://crypto.stackexchange.com/questions/111385/polynomial-time-quantum-algorithms-for-lattice-problems">this comment</a> points out: “We had already some cases where efficient quantum algorithms for lattice problems were discovered, but they turned out <a href="https://crypto.stackexchange.com/questions/41731/new-quantum-attack-on-lattices-or-shor-strikes-again?rq=1">not being correct</a> or only worked for <a href="https://crypto.stackexchange.com/questions/95187/what-does-the-work-an-efficient-quantum-algorithm-for-lattice-problems-achievin">simple special cases</a>.” I expect we’ll learn more about this particular algorithm with time. And, like many of these algorithms, there will be improvements down the road...</p>]]></description>
		
		
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		<title>In Memoriam: Ross Anderson, 1956–2024</title>
		<link>https://noise.getoto.net/2024/04/10/in-memoriam-ross-anderson-1956-2024/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Wed, 10 Apr 2024 11:08:10 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[cybersecurity]]></category>
		<category><![CDATA[economics of security]]></category>
		<category><![CDATA[security conferences]]></category>
		<category><![CDATA[security engineering]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68760</guid>

					<description><![CDATA[Last week, I posted a short memorial of Ross Anderson. The Communications of the ACM asked me to expand it. Here&#8217;s the longer version.
EDITED TO ADD (4/11): Two weeks before he passed away, Ross gave an 80-minute interview where he told his life ...]]></description>
		
		
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		<title>Ross Anderson</title>
		<link>https://noise.getoto.net/2024/04/01/ross-anderson/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Mon, 01 Apr 2024 00:21:09 +0000</pubDate>
				<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[cybersecurity]]></category>
		<category><![CDATA[economics of security]]></category>
		<category><![CDATA[security conferences]]></category>
		<category><![CDATA[security engineering]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68683</guid>

					<description><![CDATA[<p><a href="https://www.cl.cam.ac.uk/~rja14/">Ross Anderson</a> unexpectedly passed away <a href="https://www.lightbluetouchpaper.org/2024/03/29/rip-ross-anderson/">Thursday night</a> in, I believe, his home in Cambridge.</p>
<p>I can’t remember when I first met Ross. Of course it was before 2008, when we created the <a href="https://www.schneier.com/blog/archives/2023/06/security-and-human-behavior-shb-2023.html">Security and Human Behavior</a> workshop. It was well before 2001, when we created the <a href="https://econinfosec.org/">Workshop on Economics and Information Security</a>. (Okay, he created both—I helped.) It was before 1998, when we <a href="https://www.schneier.com/academic/archives/1997/04/the_risks_of_key_rec.html">wrote about</a> the problems with key escrow systems. I was one of the people he brought to the Newton Institute, at Cambridge University, for the six-month cryptography residency program he ran (I mistakenly didn’t stay the whole time)—that was in 1996...</p>]]></description>
		
		
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		<title>Apple Announces Post-Quantum Encryption Algorithms for iMessage</title>
		<link>https://noise.getoto.net/2024/02/26/apple-announces-post-quantum-encryption-algorithms-for-imessage/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Mon, 26 Feb 2024 12:04:34 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[Apple]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[encryption]]></category>
		<category><![CDATA[nist]]></category>
		<category><![CDATA[quantum computing]]></category>
		<category><![CDATA[security standards]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68476</guid>

					<description><![CDATA[<p>Apple announced <a href="https://security.apple.com/blog/imessage-pq3/">PQ3</a>, its post-quantum encryption standard based on the <a href="https://pq-crystals.org/kyber/">Kyber</a> secure key-encapsulation protocol, one of the post-quantum algorithms <a href="https://csrc.nist.gov/Projects/post-quantum-cryptography/selected-algorithms-2022">selected</a> by NIST in 2022.</p>
<p>There’s a lot of detail in the Apple <a href="https://security.apple.com/blog/imessage-pq3/">blog post</a>, and more in Douglas Stabila’s <a href="https://security.apple.com/assets/files/Security_analysis_of_the_iMessage_PQ3_protocol_Stebila.pdf">security analysis</a>.</p>
<p>I am of two minds about this. On the one hand, it’s probably premature to switch to any particular post-quantum algorithms. The mathematics of cryptanalysis for these lattice and other systems is still rapidly evolving, and we’re likely to break more of them—and learn a lot in the process—over the coming few years. But if you’re going to make the switch, this is an excellent choice. And Apple’s ability to do this so efficiently speaks well about its algorithmic agility, which is probably more important than its particular cryptographic design. And it is probably about the right time to worry about, and defend against, attackers who are storing encrypted messages in hopes of breaking them later on future quantum computers...</p>]]></description>
		
		
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		<title>Improving the Cryptanalysis of Lattice-Based Public-Key Algorithms</title>
		<link>https://noise.getoto.net/2024/02/14/improving-the-cryptanalysis-of-lattice-based-public-key-algorithms/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Wed, 14 Feb 2024 12:08:03 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[nist]]></category>
		<category><![CDATA[quantum computing]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68429</guid>

					<description><![CDATA[The winner of the Best Paper Award at Crypto this year was a significant improvement to lattice-based cryptanalysis.
This is important, because a bunch of NIST&#8217;s post-quantum options base their security on lattice problems.
I worry about standard...]]></description>
		
		
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		<item>
		<title>David Kahn</title>
		<link>https://noise.getoto.net/2024/02/02/david-kahn/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Fri, 02 Feb 2024 20:06:13 +0000</pubDate>
				<category><![CDATA[books]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[history of cryptography]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68382</guid>

					<description><![CDATA[David Kahn has died. His groundbreaking book,  The Codebreakers was the first serious book I read about codebreaking, and one of the primary reasons I entered this field.
He will be missed.
EDITED TO ADD (2/4): Funeral website.
EDITED TO ADD (2/10): Ne...]]></description>
		
		
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		<title>Improving Shor’s Algorithm</title>
		<link>https://noise.getoto.net/2024/01/05/improving-shors-algorithm/</link>
		
		<dc:creator><![CDATA[Bruce Schneier]]></dc:creator>
		<pubDate>Fri, 05 Jan 2024 12:07:35 +0000</pubDate>
				<category><![CDATA[academic papers]]></category>
		<category><![CDATA[cryptanalysis]]></category>
		<category><![CDATA[Cryptography]]></category>
		<category><![CDATA[quantum computing]]></category>
		<category><![CDATA[rsa]]></category>
		<category><![CDATA[Uncategorized]]></category>
		<guid isPermaLink="false">https://www.schneier.com/?p=68250</guid>

					<description><![CDATA[<p>We don’t have a useful quantum computer yet, but we do have quantum algorithms. Shor’s algorithm has the potential to factor large numbers faster than otherwise possible, which—if the run times are actually feasible—could break both the RSA and Diffie-Hellman public-key algorithms.</p>
<p>Now, computer scientist Oded Regev has a significant speed-up to Shor’s algorithm, at the cost of more storage.</p>
<p>Details are in <a href="https://www.quantamagazine.org/thirty-years-later-a-speed-boost-for-quantum-factoring-20231017/">this article</a>. Here’s the result:</p>
<blockquote><p>The improvement was profound. The number of elementary logical steps in the quantum part of Regev’s algorithm is proportional to ...</p></blockquote>]]></description>
		
		
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