{"id":12323,"date":"2025-02-10T17:36:23","date_gmt":"2025-02-10T17:36:23","guid":{"rendered":"https:\/\/metaschool.so\/articles\/?p=12323"},"modified":"2025-02-13T17:37:47","modified_gmt":"2025-02-13T17:37:47","slug":"ecdsa-vs-schnorr-signatures-a-battle","status":"publish","type":"post","link":"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/","title":{"rendered":"ECDSA vs. Schnorr Signatures a Battle"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_56_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title \" >Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Introduction_to_Digital_Signatures\" title=\"Introduction to Digital Signatures\">Introduction to Digital Signatures<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Mathematical_Grounding\" title=\"Mathematical Grounding\">Mathematical Grounding<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Mathematics_Behind_Elliptic_Curves\" title=\"Mathematics Behind Elliptic Curves\">Mathematics Behind Elliptic Curves<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Group_Properties\" title=\"Group Properties\">Group Properties<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Understanding_ECDSA\" title=\"Understanding ECDSA\">Understanding ECDSA<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Mathematical_Formulation_of_ECDSA\" title=\"Mathematical Formulation of ECDSA\">Mathematical Formulation of ECDSA<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Security_Proof_Sketch\" title=\"Security Proof Sketch:\">Security Proof Sketch:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#How_ECDSA_Works\" title=\"How ECDSA Works\">How ECDSA Works<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Understanding_Schnorr_Signatures\" title=\"Understanding Schnorr Signatures\">Understanding Schnorr Signatures<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Mathematical_Formulation_of_Schnorr_Signatures\" title=\"Mathematical Formulation of Schnorr Signatures\">Mathematical Formulation of Schnorr Signatures<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Security_Proof\" title=\"Security Proof:\">Security Proof:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#The_Schnorr_Advantage\" title=\"The Schnorr Advantage\">The Schnorr Advantage<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Detailed_Comparison\" title=\"Detailed Comparison\">Detailed Comparison<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Key_Differences_and_Practical_Implications\" title=\"Key Differences and Practical Implications\">Key Differences and Practical Implications<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Security\" title=\"Security\">Security<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Efficiency\" title=\"Efficiency\">Efficiency<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Implementation\" title=\"Implementation\">Implementation<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Implementation_Examples\" title=\"Implementation Examples\">Implementation Examples<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#ECDSA_Implementation\" title=\"ECDSA Implementation\">ECDSA Implementation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Schnorr_Implementation\" title=\"Schnorr Implementation\">Schnorr Implementation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Batch_Verification_Example\" title=\"Batch Verification Example\">Batch Verification Example<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Implementation_Considerations\" title=\"Implementation Considerations\">Implementation Considerations<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Which_One_Should_You_Use\" title=\"Which One Should You Use?\">Which One Should You Use?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Looking_to_the_Future\" title=\"Looking to the Future\">Looking to the Future<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/metaschool.so\/articles\/ecdsa-vs-schnorr-signatures-a-battle\/#Conclusion\" title=\"Conclusion\">Conclusion<\/a><\/li><\/ul><\/nav><\/div>\n\n<p>Digital signatures represent the backbone of modern cryptographic systems, serving as the building blocks for secure communication, blockchain transactions, and even verification of digital identity. We are going to examine two such digital signature algorithms, namely, the wide-use Elliptic Curve Digital Signature Algorithm (ECDSA) and the newer gaining popularity Schnorr signature scheme.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Introduction_to_Digital_Signatures\"><\/span>Introduction to Digital Signatures<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Consider sending a letter in the physical world. You sign the letter with your hand-written signature to prove that it came from you. In the digital world, digital signatures serve the same purpose but with mathematical precision and cryptographic security. They provide three essential properties for messages:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Authentication: Proving who created or sent a message<\/li>\n\n\n\n<li>Non-repudiation: The sender cannot deny sending the message<\/li>\n\n\n\n<li>Integrity: Verifying the message has not been tampered with<\/li>\n<\/ol>\n\n\n\n<p>These attributes have made digital signatures quintessential for anything ranging from cryptocurrency transactions and authentication of software updates to the security of messaging systems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mathematical_Grounding\"><\/span>Mathematical Grounding<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Let&#8217;s briefly introduce the mathematics underlying both ECDSA and Schnorr signatures.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mathematics_Behind_Elliptic_Curves\"><\/span>Mathematics Behind Elliptic Curves<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The basis of both schemes consists of elliptic curves described by the equation:<\/p>\n\n\n\n<p>y\u00b2 = x\u00b3 + ax + b (mod p)<\/p>\n\n\n\n<p>with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a and b are curve parameters<\/li>\n\n\n\n<li>Points on this curve form a cyclic group under point addition<\/li>\n<\/ul>\n\n\n\n<p>Important Mathematical Properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Point Addition: P + Q = R where P, Q and R are points on the curve<\/li>\n\n\n\n<li>Scalar Multiplication: kP = P + P +. + P (k times)<\/li>\n\n\n\n<li>Discrete Log Problem: Given P and Q = kP then it is computationally infeasible to find k<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Group_Properties\"><\/span>Group Properties<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Both schemes are based on the following:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A cyclic group G of prime order n<\/li>\n\n\n\n<li>A generator point G<\/li>\n\n\n\n<li>A point at infinity O (neutral element)<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Understanding_ECDSA\"><\/span>Understanding ECDSA<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Over the years, ECDSA has been the workhorse of digital signatures in blockchain and other cryptographic systems. It is an extension from the mathematical properties of elliptic curves toward producing secure signatures that are relatively small in size compared to other systems such as RSA.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mathematical_Formulation_of_ECDSA\"><\/span>Mathematical Formulation of ECDSA<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Key Generation:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Private key: Random integer d \u2208 [1, n-1]<\/li>\n\n\n\n<li>Public key: Q = dG<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Signature Generation:<br>For a message m:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Generate random k \u2208 [1, n-1]<\/li>\n\n\n\n<li>Compute R = kG = (x\u2081, y\u2081)<\/li>\n\n\n\n<li>r = x\u2081 mod n<\/li>\n\n\n\n<li>s = k\u207b\u00b9(H(m) + dr) mod n<\/li>\n\n\n\n<li>Signature is: (r, s)<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Signature Verification:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Compute w = s\u207b\u00b9 mod n<\/li>\n\n\n\n<li>u\u2081 = H(m)w mod n<\/li>\n\n\n\n<li>u\u2082 = rw mod n<\/li>\n\n\n\n<li>Compute point: (x\u2081, y\u2081) = u\u2081G + u\u2082Q<\/li>\n\n\n\n<li>Check r = x\u2081 mod n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Security_Proof_Sketch\"><\/span>Security Proof Sketch:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The security of ECDSA is based on the:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The elliptic curve group discrete logarithm problem<\/li>\n\n\n\n<li>The random oracle model of hash function H<\/li>\n\n\n\n<li>The signature scheme security against chosen message attack<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"How_ECDSA_Works\"><\/span>How ECDSA Works<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>In the signature scheme of ECDSA, the signature process is achieved as follows:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Key Generation<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Generate a random private key (a large number)<\/li>\n\n\n\n<li>This private key multiplied by the generator point on the elliptic curve yields the public key<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Signing<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Hash of a message<\/li>\n\n\n\n<li>Generate a random number (nonce)<\/li>\n\n\n\n<li>Create the signature from the combination of message hash, private key, and nonce<\/li>\n\n\n\n<li>The signature would consist of two values: r and s<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Verification<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Using the public key, message hash, and signature values<\/li>\n\n\n\n<li>Perform elliptic curve operations to verify the signature&#8217;s validity<\/li>\n<\/ul>\n\n\n\n<p>ECDSA&#8217;s strength lies in its widespread adoption and battle-tested implementation. Bitcoin and Ethereum both traditionally use ECDSA for transaction signatures. However, it does have some limitations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Nonce Reuse: Using the same nonce for different signatures can compromise the private key<\/li>\n\n\n\n<li>Signature Malleability: The signature can be slightly altered and still be valid<\/li>\n\n\n\n<li>Complex Multi-signature Implementation: Combining multiple signatures is not straightforward<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Understanding_Schnorr_Signatures\"><\/span>Understanding Schnorr Signatures<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Schnorr signatures, though actually invented before it, have recently gained renewed attention due to their elegant mathematical properties and some very practical advantages. They were initially patented, which limited their widespread adoption, but are now freely available and find increasing implementation in modern systems.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mathematical_Formulation_of_Schnorr_Signatures\"><\/span>Mathematical Formulation of Schnorr Signatures<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Key Generation:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Private key: Random integer x \u2208 [1, n-1]<\/li>\n\n\n\n<li>Public key: P = xG<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Signature Generation:<br>For a message m:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Generate random k \u2208 [1, n-1]<\/li>\n\n\n\n<li>Calculate R = kG<\/li>\n\n\n\n<li>Calculate e = H(R || P || m)<\/li>\n\n\n\n<li>Calculate s = k + ex mod n<\/li>\n\n\n\n<li>Signature is: (R, s)<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Signature Verification:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Compute: e = H(R || P || m)<\/li>\n\n\n\n<li>Check: sG = R + eP<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Security_Proof\"><\/span>Security Proof:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Schnorr signatures provide stronger security guarantees through:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Provable security in the algebraic group model<\/li>\n\n\n\n<li>Reduction to the discrete logarithm problem<\/li>\n\n\n\n<li>Linear construction enabling easier security proofs<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"The_Schnorr_Advantage\"><\/span>The Schnorr Advantage<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Schnorr signatures work similarly to ECDSA but with some key improvements:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Simpler Mathematics<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Linear relationship between components makes it easier to prove security<\/li>\n\n\n\n<li>More straightforward implementation reduces the risk of bugs<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Native Multi-signature Support<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Multiple signatures can be aggregated into a single signature<\/li>\n\n\n\n<li>This saves space and improves privacy in blockchain applications<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Batch Verification<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>More signatures are verified much faster individually<\/li>\n\n\n\n<li>Particularly useful for verification on blockchain nodes that have to verify plenty of signatures<\/li>\n<\/ul>\n\n\n\n<p>Recently, the Bitcoin network adopted the Schnorr signature with the Taproot upgrade for efficient and private smart contract execution.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Detailed_Comparison\"><\/span>Detailed Comparison<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Feature<\/th><th>ECDSA<\/th><th>Schnorr Signatures<\/th><\/tr><\/thead><tbody><tr><td><strong>Mathematical Properties<\/strong><\/td><td><\/td><td><\/td><\/tr><tr><td>Security Basis<\/td><td>Discrete Log Problem<\/td><td>Discrete Log Problem<\/td><\/tr><tr><td>Signature Size<\/td><td>64 bytes (r,s)<\/td><td>64 bytes (R,s)<\/td><\/tr><tr><td>Linearity<\/td><td>Non-linear signature equation<\/td><td>Linear signature equation<\/td><\/tr><tr><td>Provable Security<\/td><td>Complex security proof<\/td><td>Simpler security proof<\/td><\/tr><tr><td><strong>Implementation Characteristics<\/strong><\/td><td><\/td><td><\/td><\/tr><tr><td>Nonce Requirements<\/td><td>Strict (reuse breaks security)<\/td><td>Strict (reuse breaks security)<\/td><\/tr><tr><td>Implementation Complexity<\/td><td>More complex<\/td><td>Simpler<\/td><\/tr><tr><td>Side-Channel Attack Resistance<\/td><td>Requires careful implementation<\/td><td>Better inherent resistance<\/td><\/tr><tr><td><strong>Advanced Features<\/strong><\/td><td><\/td><td><\/td><\/tr><tr><td>Multi-signature Support<\/td><td>Complex to implement<\/td><td>Native support<\/td><\/tr><tr><td>Signature Aggregation<\/td><td>Limited<\/td><td>Efficient<\/td><\/tr><tr><td>Batch Verification<\/td><td>Less efficient<\/td><td>More efficient<\/td><\/tr><tr><td><strong>Practical Aspects<\/strong><\/td><td><\/td><td><\/td><\/tr><tr><td>Adoption Level<\/td><td>Widely adopted<\/td><td>Growing adoption<\/td><\/tr><tr><td>Standardization<\/td><td>Well-standardized<\/td><td>Newer standards emerging<\/td><\/tr><tr><td>Compatibility<\/td><td>Extensive ecosystem<\/td><td>Limited but growing<\/td><\/tr><tr><td><strong>Performance<\/strong><\/td><td><\/td><td><\/td><\/tr><tr><td>Verification Speed<\/td><td>Base performance<\/td><td>~15% faster<\/td><\/tr><tr><td>Signature Generation Speed<\/td><td>Base performance<\/td><td>Similar<\/td><\/tr><tr><td>Memory Requirements<\/td><td>Base requirement<\/td><td>Slightly lower<\/td><\/tr><tr><td><strong>Security Properties<\/strong><\/td><td><\/td><td><\/td><\/tr><tr><td>Malleability<\/td><td>Malleable<\/td><td>Non-malleable<\/td><\/tr><tr><td>Key Prefixing Requirements<\/td><td>Required for security<\/td><td>Not required<\/td><\/tr><tr><td>Replay Attack Protection<\/td><td>Requires additional measures<\/td><td>Built-in protection<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Key_Differences_and_Practical_Implications\"><\/span>Key Differences and Practical Implications<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Looking into the differences between ECDSA and Schnorr signatures, a few key factors come into play:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"796\" height=\"1024\" src=\"https:\/\/metaschool.so\/articles\/wp-content\/uploads\/2025\/02\/freepik__an-educational-illustration-explaining-the-mathema__60122-796x1024.png\" alt=\"ECDSA vs Schnorr\n\" class=\"wp-image-12392\" srcset=\"https:\/\/metaschool.so\/articles\/wp-content\/uploads\/2025\/02\/freepik__an-educational-illustration-explaining-the-mathema__60122-796x1024.png 796w, https:\/\/metaschool.so\/articles\/wp-content\/uploads\/2025\/02\/freepik__an-educational-illustration-explaining-the-mathema__60122-233x300.png 233w, https:\/\/metaschool.so\/articles\/wp-content\/uploads\/2025\/02\/freepik__an-educational-illustration-explaining-the-mathema__60122-768x987.png 768w, https:\/\/metaschool.so\/articles\/wp-content\/uploads\/2025\/02\/freepik__an-educational-illustration-explaining-the-mathema__60122.png 896w\" sizes=\"auto, (max-width: 796px) 100vw, 796px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Security\"><\/span>Security<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Schnorr signatures<\/strong> provide provable security based on standard cryptographic assumptions<\/li>\n\n\n\n<li><strong>ECDSA<\/strong> has more complicated security proofs, depending on stronger assumptions<\/li>\n\n\n\n<li>For practical applications today, both are secure<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Efficiency\"><\/span>Efficiency<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Schnorr signatures are typically smaller in size<\/li>\n\n\n\n<li>Verification of Schnorr signatures can be batched more efficiently<\/li>\n\n\n\n<li>ECDSA requires more complex calculations for signature verification<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Implementation\"><\/span>Implementation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>ECDSA is more widely supported in existing systems<\/li>\n\n\n\n<li>Schnorr signatures are easier to implement correctly<\/li>\n\n\n\n<li>Schnorr&#8217;s linear nature makes it more suitable for advanced cryptographic protocols<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Implementation_Examples\"><\/span>Implementation Examples<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ECDSA_Implementation\"><\/span>ECDSA Implementation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Here&#8217;s a practical implementation of ECDSA using Python&#8217;s <code>fastecdsa<\/code> library:<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"from fastecdsa import keys, curve, ecdsa\nfrom hashlib import sha256\n\ndef generate_ecdsa_keypair():\n    &quot;&quot;&quot;Generate ECDSA key pair.&quot;&quot;&quot;\n    private_key = keys.gen_private_key(curve.secp256k1)\n    public_key = keys.get_public_key(private_key, curve.secp256k1)\n    return private_key, public_key\n\ndef sign_ecdsa(message: str, private_key: int) -&gt; tuple:\n    &quot;&quot;&quot;\n    Sign a message using ECDSA.\n\n    Args:\nsignature The message to be signed\n        private_key - The private key for signing\n        #       Returns:\n        tuple -      (r,s) signature parts\n    &quot;&quot;&quot;&quot;&quot;&quot;\n    message_hash = int(sha256(message.encode('utf-8')).hexdigest(), 16)\nr, s = ecdsa.sign(message_hash, private_key, curve=curve.secp256k1)\nreturn r, s\ndef verify_ecdsa(message: str, signature: tuple, public_key) -&gt; bool:\n    &quot;&quot;&quot;&quot;\n    Verify an ECDSA signature.\n\n    Args:\n        message: The original message\n        signature: (r, s) signature components\npublic_key: The public key for verification\n\n    Returns:\n        bool: True if signature is valid, False otherwise\n    &quot;&quot;&quot;\n    message_hash = int(sha256(message.encode('utf-8')).hexdigest(), 16)\n    r, s = signature\n    return ecdsa.verify(signature, message_hash, public_key, curve=curve.secp256k1)\n\n# Example usage\ndef ecdsa_example():\n    # Generate keypair\n    private_key, public_key = generate_ecdsa_keypair()\n\n    # Sign a message\n    message = &quot;Hello, Blockchain!&quot;\n    signature = sign_ecdsa(message, private_key)\n\n    # Verify the signature\n    is_valid = verify_ecdsa(message, signature, public_key)\n    return is_valid\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #9CDCFE\">from<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">fastecdsa<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">import<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">keys<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">ecdsa<\/span><\/span>\n<span class=\"line\"><span style=\"color: #C586C0\">from<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">hashlib<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">import<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">sha256<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">generate_ecdsa_keypair<\/span><span style=\"color: #D4D4D4\">():<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;Generate ECDSA key pair.&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">keys<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">gen_private_key<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">keys<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">get_public_key<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">sign_ecdsa<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">str<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #9CDCFE\">tuple<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">    Sign a message using ECDSA<\/span><span style=\"color: #F44747\">.<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">Args<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">The<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">to<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">be<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signed<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\"> - <\/span><span style=\"color: #9CDCFE\">The<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">private<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">key<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signing<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        #       <\/span><span style=\"color: #9CDCFE\">Returns<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">tuple<\/span><span style=\"color: #D4D4D4\"> -      (<\/span><span style=\"color: #9CDCFE\">r<\/span><span style=\"color: #D4D4D4\">,<\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">parts<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">message_hash<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">sha256<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">encode<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #CE9178\">&#39;utf-8&#39;<\/span><span style=\"color: #D4D4D4\">)).<\/span><span style=\"color: #9CDCFE\">hexdigest<\/span><span style=\"color: #D4D4D4\">(), 16)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">r<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">ecdsa<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">sign<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message_hash<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">=<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">r<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">verify_ecdsa<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">str<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">tuple<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #9CDCFE\">bool<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">Verify<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">an<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">ECDSA<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">.<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">Args<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">The<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">original<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">message<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">: (<\/span><span style=\"color: #9CDCFE\">r<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">components<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">The<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">public<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">key<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">verification<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">Returns<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">bool<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">True<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">if<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">is<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">valid<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">False<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">otherwise<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">    message_hash = int(sha256(message.encode(&#39;utf-8&#39;)).hexdigest(), 16<\/span><span style=\"color: #F44747\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">r<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">signature<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">ecdsa<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">verify<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">message_hash<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">=<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\"># <\/span><span style=\"color: #9CDCFE\">Example<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">usage<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">ecdsa_example<\/span><span style=\"color: #D4D4D4\">():<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Generate<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">keypair<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">generate_ecdsa_keypair<\/span><span style=\"color: #D4D4D4\">()<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Sign<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">a<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">message<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #CE9178\">&quot;Hello, Blockchain!&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">sign_ecdsa<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Verify<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">the<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">is_valid<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">verify_ecdsa<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">is_valid<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Schnorr_Implementation\"><\/span>Schnorr Implementation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Here is an example implementation of Schnorr signatures in Python. For ease of understanding, this is implemented in a simplified manner and not for production use:<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"from hashlib import sha256\nimport random\nfrom fastecdsa.curve import secp256k1\nfrom fastecdsa.point import Point\n\nclass SchnorrSignature:\n    def __init__(self):\n        self.curve = secp256k1\n        self.G = self.curve.G\n        self.n = self.curve.q\n\n    def generate_keypair(self):\n&quot;&quot;&quot;Generate Schnorr key pair.&quot;&quot;&quot;\n        private_key = random.randrange(1, self.n)\n        public_key = private_key * self.G\n        return private_key, public_key\n\n    def hash_point(self, point: Point) -&gt; int:\n        &quot;&quot;&quot;Hash an elliptic curve point.&quot;&quot;&quot;\nreturn int(sha256(f&quot;{point.x},{point.y}&quot;.encode()).hexdigest(), 16)\n\n    def hash_message(self, R: Point, P: Point, message: str) -&gt; int:\n        &quot;&quot;&quot;Hash the commitment, public key and message.&quot;&quot;&quot;\n        message_bytes = message.encode('utf-8')\n        hash_input = f&quot;{R.x},{R.y},{P.x},{P.y},{message_bytes.hex()}&quot;.encode()\nreturn int(sha256(hash_input).hexdigest(), 16) % self.n\n\n    def sign(self, message: str, private_key: int, public_key: Point) -&gt; tuple:\n        &quot;&quot;&quot;\n        Create a Schnorr signature.\n\n        Args:\nmessage: Message to sign\n            private_key: Private key\n            public_key: Public key (P = x*G)\n\n        Returns:\ntuple: (R, s) signature components\n        &quot;&quot;&quot;\n        # Generate random nonce\n        k = random.randrange(1, self.n)\n        R = k * self.G\n# Calculate challenge\n        e = self.hash_message(R, public_key, message)\n\n        # Calculate signature\n        s = (k + e * private_key) % self.n\nreturn (R, s)\n\n    def verify(self, message: str, signature: tuple, public_key: Point) -&gt; bool:\n        &quot;&quot;&quot;\n        Verify a Schnorr signature.\n\n        Args:\nmessage: Original message\n            signature: (R, s) signature components\n            public_key: Public key for verification\n\n        Returns:\nbool: True if signature is valid\n        &quot;&quot;&quot;\n        R, s = signature\n\n        # Compute challenge\n        e = self.hash_message(R, public_key, message)\n# Verify signature equation: sG = R + eP\n        left_side = s * self.G\n        right_side = R + e * public_key\nreturn left_side == right_side\n\n# Example usage\ndef schnorr_example():\n    schnorr = SchnorrSignature()\n\n    # Generate keypair\n    private_key, public_key = schnorr.generate_keypair()\n\n    # Sign a message\n    message = &quot;Hello, Blockchain!&quot;\n    signature = schnorr.sign(message, private_key, public_key)\n# Verify the signature\n    is_valid = schnorr.verify(message, signature, public_key)\n    return is_valid\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #9CDCFE\">from<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">hashlib<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">import<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">sha256<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">import<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">random<\/span><\/span>\n<span class=\"line\"><span style=\"color: #C586C0\">from<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">fastecdsa<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">import<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><\/span>\n<span class=\"line\"><span style=\"color: #C586C0\">from<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">fastecdsa<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">point<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">import<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">Point<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">class<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">SchnorrSignature<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">__init__<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">G<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">G<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">q<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">generate_keypair<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">&quot;&quot;&quot;Generate Schnorr key pair.&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">random<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">randrange<\/span><span style=\"color: #D4D4D4\">(1, <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">*<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">G<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">hash_point<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">point<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Point<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;Hash an elliptic curve point.&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">sha256<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">f<\/span><span style=\"color: #CE9178\">&quot;{point.x},{point.y}&quot;<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">encode<\/span><span style=\"color: #D4D4D4\">()).<\/span><span style=\"color: #9CDCFE\">hexdigest<\/span><span style=\"color: #D4D4D4\">(), 16)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">hash_message<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Point<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">P<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Point<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">str<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;Hash the commitment, public key and message.&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">message_bytes<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">encode<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #CE9178\">&#39;utf-8&#39;<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">hash_input<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">f<\/span><span style=\"color: #CE9178\">&quot;{R.x},{R.y},{P.x},{P.y},{message_bytes.hex()}&quot;<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">encode<\/span><span style=\"color: #D4D4D4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">sha256<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">hash_input<\/span><span style=\"color: #D4D4D4\">).<\/span><span style=\"color: #9CDCFE\">hexdigest<\/span><span style=\"color: #D4D4D4\">(), 16) % <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">sign<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">str<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">int<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Point<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #9CDCFE\">tuple<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">        Create a Schnorr signature<\/span><span style=\"color: #F44747\">.<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">Args<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Message<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">to<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">sign<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Private<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">key<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Public<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">key<\/span><span style=\"color: #D4D4D4\"> (<\/span><span style=\"color: #9CDCFE\">P<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">x<\/span><span style=\"color: #569CD6\">*<\/span><span style=\"color: #9CDCFE\">G<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">Returns<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">tuple<\/span><span style=\"color: #D4D4D4\">: (<\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">components<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">        # Generate random nonc<\/span><span style=\"color: #F44747\">e<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">k<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">random<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">randrange<\/span><span style=\"color: #D4D4D4\">(1, <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">k<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">*<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">G<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\"># <\/span><span style=\"color: #9CDCFE\">Calculate<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">challenge<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">e<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">hash_message<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        # <\/span><span style=\"color: #9CDCFE\">Calculate<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\"> = (<\/span><span style=\"color: #9CDCFE\">k<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #9CDCFE\">e<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">*<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">) % <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> (<\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">verify<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">str<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">tuple<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Point<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #9CDCFE\">bool<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">        Verify a Schnorr signature<\/span><span style=\"color: #F44747\">.<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">Args<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Original<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">message<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">: (<\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">components<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">            <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">Public<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">key<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">verification<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">Returns<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">bool<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">True<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">if<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">is<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">valid<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">        R, s = signatur<\/span><span style=\"color: #F44747\">e<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        # <\/span><span style=\"color: #9CDCFE\">Compute<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">challenge<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">e<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">hash_message<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\"># <\/span><span style=\"color: #9CDCFE\">Verify<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">equation<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">sG<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #9CDCFE\">eP<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">left_side<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">*<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">self<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">G<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">right_side<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">R<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #9CDCFE\">e<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">*<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">public_key<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">left_side<\/span><span style=\"color: #D4D4D4\"> == <\/span><span style=\"color: #9CDCFE\">right_side<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\"># <\/span><span style=\"color: #9CDCFE\">Example<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">usage<\/span><\/span>\n<span class=\"line\"><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">schnorr_example<\/span><span style=\"color: #D4D4D4\">():<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">SchnorrSignature<\/span><span style=\"color: #D4D4D4\">()<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Generate<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">keypair<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">generate_keypair<\/span><span style=\"color: #D4D4D4\">()<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Sign<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">a<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">message<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #CE9178\">&quot;Hello, Blockchain!&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">sign<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">private_key<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\"># <\/span><span style=\"color: #9CDCFE\">Verify<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">the<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signature<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">is_valid<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">verify<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">message<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signature<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_key<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">is_valid<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Batch_Verification_Example\"><\/span>Batch Verification Example<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Below is an example of how Schnorr signatures enable efficient batch verification:<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#1E1E1E\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" data-code=\"def batch_verify_schnorr(messages: list, signatures: list, public_keys: list) -&gt; bool:\n    &quot;&quot;&quot;\n    Batch verify multiple Schnorr signatures.\n    \n    Args:\n        messages: List of messages\n        signatures: List of signatures (R, s)\n        public_keys: List of public keys\n        \n    Returns:\n        bool: True if all signatures are valid\n    &quot;&quot;&quot;\n    schnorr = SchnorrSignature()\n    n = len(messages)\n    \n    # Generate random weights\n    weights = [random.randrange(1, schnorr.n) for _ in range(n)]\n    \n    # Combine signatures\n    combined_s = sum(w * s for w, (_, s) in zip(weights, signatures)) % schnorr.n\n    combined_R = sum(w * R for w, (R, _) in zip(weights, signatures))\n    \n    # Combine public keys and challenges\n    combined_P = Point(0, 0, curve=secp256k1)  # Identity point\n    for i in range(n):\n        e = schnorr.hash_message(signatures[i][0], public_keys[i], messages[i])\n        combined_P = combined_P + (weights[i] * e * public_keys[i])\n    \n    # Verify batch\n    return combined_s * schnorr.G == combined_R + combined_P\" style=\"color:#D4D4D4;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki dark-plus\" style=\"background-color: #1E1E1E\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #9CDCFE\">def<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">batch_verify_schnorr<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">messages<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">list<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signatures<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">list<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">public_keys<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">list<\/span><span style=\"color: #D4D4D4\">) -&gt; <\/span><span style=\"color: #C8C8C8\">bool<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">    Batch verify multiple Schnorr signatures<\/span><span style=\"color: #F44747\">.<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #C8C8C8\">Args<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C8C8C8\">messages<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">List<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">of<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">messages<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C8C8C8\">signatures<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">List<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">of<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">signatures<\/span><span style=\"color: #D4D4D4\"> (<\/span><span style=\"color: #4FC1FF\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C8C8C8\">public_keys<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">List<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">of<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">public<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">keys<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #C8C8C8\">Returns<\/span><span style=\"color: #D4D4D4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #C8C8C8\">bool<\/span><span style=\"color: #D4D4D4\">: <\/span><span style=\"color: #9CDCFE\">True<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #C586C0\">if<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">all<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signatures<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">are<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">valid<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #CE9178\">&quot;&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #CE9178\">    schnorr = SchnorrSignature(<\/span><span style=\"color: #F44747\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #DCDCAA\">len<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">messages<\/span><span style=\"color: #D4D4D4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Generate<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">random<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">weights<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">weights<\/span><span style=\"color: #D4D4D4\"> = [<\/span><span style=\"color: #9CDCFE\">random<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #DCDCAA\">randrange<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #B5CEA8\">1<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">_<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\">)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Combine<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">signatures<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">combined_s<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">w<\/span><span style=\"color: #D4D4D4\"> * <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">w<\/span><span style=\"color: #D4D4D4\">, (<\/span><span style=\"color: #9CDCFE\">_<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">s<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #569CD6\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">zip<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">weights<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signatures<\/span><span style=\"color: #D4D4D4\">)) % <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #9CDCFE\">n<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">combined_R<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #DCDCAA\">sum<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">w<\/span><span style=\"color: #D4D4D4\"> * <\/span><span style=\"color: #4FC1FF\">R<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">w<\/span><span style=\"color: #D4D4D4\">, (<\/span><span style=\"color: #4FC1FF\">R<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">_<\/span><span style=\"color: #D4D4D4\">) <\/span><span style=\"color: #569CD6\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">zip<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">weights<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">signatures<\/span><span style=\"color: #D4D4D4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Combine<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">public<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">keys<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">and<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">challenges<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">combined_P<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #DCDCAA\">Point<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">, <\/span><span style=\"color: #9CDCFE\">curve<\/span><span style=\"color: #D4D4D4\">=<\/span><span style=\"color: #9CDCFE\">secp256k1<\/span><span style=\"color: #D4D4D4\">)  # <\/span><span style=\"color: #9CDCFE\">Identity<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">point<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #9CDCFE\">for<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">i<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #569CD6\">in<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #DCDCAA\">range<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">n<\/span><span style=\"color: #D4D4D4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">e<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #DCDCAA\">hash_message<\/span><span style=\"color: #D4D4D4\">(<\/span><span style=\"color: #9CDCFE\">signatures<\/span><span style=\"color: #D4D4D4\">[<\/span><span style=\"color: #9CDCFE\">i<\/span><span style=\"color: #D4D4D4\">][<\/span><span style=\"color: #B5CEA8\">0<\/span><span style=\"color: #D4D4D4\">], <\/span><span style=\"color: #9CDCFE\">public_keys<\/span><span style=\"color: #D4D4D4\">[<\/span><span style=\"color: #9CDCFE\">i<\/span><span style=\"color: #D4D4D4\">], <\/span><span style=\"color: #9CDCFE\">messages<\/span><span style=\"color: #D4D4D4\">[<\/span><span style=\"color: #9CDCFE\">i<\/span><span style=\"color: #D4D4D4\">])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">        <\/span><span style=\"color: #9CDCFE\">combined_P<\/span><span style=\"color: #D4D4D4\"> = <\/span><span style=\"color: #9CDCFE\">combined_P<\/span><span style=\"color: #D4D4D4\"> + (<\/span><span style=\"color: #9CDCFE\">weights<\/span><span style=\"color: #D4D4D4\">[<\/span><span style=\"color: #9CDCFE\">i<\/span><span style=\"color: #D4D4D4\">] * <\/span><span style=\"color: #9CDCFE\">e<\/span><span style=\"color: #D4D4D4\"> * <\/span><span style=\"color: #9CDCFE\">public_keys<\/span><span style=\"color: #D4D4D4\">[<\/span><span style=\"color: #9CDCFE\">i<\/span><span style=\"color: #D4D4D4\">])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    # <\/span><span style=\"color: #9CDCFE\">Verify<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">batch<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D4D4D4\">    <\/span><span style=\"color: #C586C0\">return<\/span><span style=\"color: #D4D4D4\"> <\/span><span style=\"color: #9CDCFE\">combined_s<\/span><span style=\"color: #D4D4D4\"> * <\/span><span style=\"color: #9CDCFE\">schnorr<\/span><span style=\"color: #D4D4D4\">.<\/span><span style=\"color: #4FC1FF\">G<\/span><span style=\"color: #D4D4D4\"> == <\/span><span style=\"color: #9CDCFE\">combined_R<\/span><span style=\"color: #D4D4D4\"> + <\/span><span style=\"color: #9CDCFE\">combined_P<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Implementation_Considerations\"><\/span>Implementation Considerations<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>While implementing these signature schemes in production, consider the following:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Secure Random Number Generation<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use cryptographically secure random number generators<\/li>\n\n\n\n<li>Never reuse nonces (k values)<\/li>\n\n\n\n<li>Implement deterministic nonce generation RFC 6979<\/li>\n<\/ul>\n\n\n\n<p>2. <strong>Side Channel Attack Prevention<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use constant time operations<\/li>\n\n\n\n<li>Avoid conditionals based on secret values<\/li>\n\n\n\n<li>Use secure memory handling<\/li>\n<\/ul>\n\n\n\n<p>3. <strong>Error Handling<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always validate inputs<\/li>\n\n\n\n<li>Handle edge cases well (infinity points, zero values)<\/li>\n\n\n\n<li>Instead use appropriate error handling.<\/li>\n<\/ul>\n\n\n\n<p>4. <strong>Performance Optimization<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Whenever possible, use optimized libraries for curve operations.<\/li>\n\n\n\n<li>When appropriate, apply batching verification.<\/li>\n\n\n\n<li>If highly performant on a large scale, use hardware acceleration.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Which_One_Should_You_Use\"><\/span>Which One Should You Use?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Use one or the other according to your specific requirements:<\/p>\n\n\n\n<p>ECDSA if you require:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>wide compatibility with most of the current systems<\/li>\n\n\n\n<li>You are working with established blockchain platforms that require ECDSA<\/li>\n\n\n\n<li>Your implementation needs to be immediately compatible with existing tools<\/li>\n<\/ul>\n\n\n\n<p>Choose Schnorr signatures if:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>You&#8217;re building a new system with multi-signature requirements<\/li>\n\n\n\n<li>Efficiency and signature size are critical concerns<\/li>\n\n\n\n<li>You need advanced features like signature aggregation<\/li>\n\n\n\n<li>You&#8217;re implementing new blockchain or cryptographic protocols<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Looking_to_the_Future\"><\/span>Looking to the Future<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>The cryptographic community increasingly favors Schnorr signatures for new implementations due to the following reasons:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Greater traction within large blockchain networks<\/li>\n\n\n\n<li>A general trend toward better privacy and efficiency in cryptographic schemes<\/li>\n\n\n\n<li>The requirement for more flexible signature schemes in advanced protocols<\/li>\n<\/ul>\n\n\n\n<p>However, ECDSA will still be relevant due to its extensive deployment and continued support by major systems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Both of them are powerful cryptographic tools that have their merits. While ECDSA has been the backbone of many cryptographic systems, Schnorr signatures are the future, with their elegant design and advanced features. Moving forward, we are likely to see increased adoption of Schnorr signatures, especially in new systems and protocols that can benefit from the advantages they bring.<\/p>\n\n\n\n<p>In this case, the choice between them should be guided by your specific needs: compatibility with existing systems, efficiency requirements, and the need for advanced features like signature aggregation. Understanding both algorithms and their trade-offs enables you to make informed decisions in your cryptographic implementations.<\/p>\n\n\n\n<p>Remember, cryptographic security is a factor not only in the choice of algorithm, but also in its proper implementation and correct key management, along with system design as a whole. Follow best practices and, when possible, take the advice of security experts for production deployment.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":24,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"neve_meta_sidebar":"","neve_meta_container":"","neve_meta_enable_content_width":"","neve_meta_content_width":0,"neve_meta_title_alignment":"","neve_meta_author_avatar":"","neve_post_elements_order":"","neve_meta_disable_header":"","neve_meta_disable_footer":"","neve_meta_disable_title":"","footnotes":""},"categories":[14],"tags":[],"class_list":["post-12323","post","type-post","status-publish","format-standard","hentry","category-web3"],"_links":{"self":[{"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/posts\/12323","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/users\/24"}],"replies":[{"embeddable":true,"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/comments?post=12323"}],"version-history":[{"count":3,"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/posts\/12323\/revisions"}],"predecessor-version":[{"id":12393,"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/posts\/12323\/revisions\/12393"}],"wp:attachment":[{"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/media?parent=12323"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/categories?post=12323"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/metaschool.so\/articles\/wp-json\/wp\/v2\/tags?post=12323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}