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### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > > POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > >### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code!### SHANKS & TONELLI

Shanks & Tonelli Pseudocode for the method of Shanks and Tonelli for the computation of square roots in respect to prime moduli is### presented below.

PRIME CURVE JACOBIAN COORDINATES Prime Curve Jacobian Coordinates Introduction 'Jacobian Coordinates' are used to represent elliptic curve points on prime curves y^2 = x^3### + ax + b.

PRIME CURVE CHUDNOVSKY COORDINATES Prime Curve Chudnovsky Coordinates Introduction 'Chudnovsky Coordinates' are used to represent elliptic curve points on prime curves y^2 = x^3 + ax + b. PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > >### LEGENDRE SYMBOL

Legendre Symbol The Legendre symbol (a|p) tells us, whether a is a square modulo a prime p or not. Here is some Python code to compute it: def legendre(a, p): if a == 0: return 0 x, y, L = a, p, 1 while 1: if x > (y >> 1): x = y - x if y & 3 == 3: L = -L while x & 3 == 0: x = x >> 2 if x & 1 == 0: x = x >> 1 if y & 7 == 3 or y & 7 == 5: L = -L if x == 1: return L if x & 3 == 3 and y & 3 == 3 POINT-AT-INFINITY.ORG Created Date: 2/4/2004 11:19:00 PM### INDEX OF PICTURES

Index of Pictures : created with igaligal POINT-AT-INFINITY.ORG Created Date: 2/13/2003 11:47:00 AM POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > > POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > >### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. POINT-AT-INFINITY.ORG http://point-at-infinity.org/ecc/nisttv Test vectors for the NIST elliptic curves P192, P224, P256, P384, P521, B163, B233, B283, B409, B571, K163, K233, K283, K409### SHANKS & TONELLI

Shanks & Tonelli Pseudocode for the method of Shanks and Tonelli for the computation of square roots in respect to prime moduli is### presented below.

PRIME CURVE JACOBIAN COORDINATES Prime Curve Jacobian Coordinates Introduction 'Jacobian Coordinates' are used to represent elliptic curve points on prime curves y^2 = x^3### + ax + b.

### MODULAR INVERSION

Modular Inversion Python code for two algorithms to compute the inverse of an arbitrary element of a prime field is presented below. The first is an implementation of algorithm 2.107 in the Handbook of Applied Cryptography. def inverse_mod(x, p): assert 0 x p a, b, x2, x1, y2, y1 = p, x, 1, 0, 0, 1 while b > 0: , a = divmod(a, b), b x1, x2 = x2 - q * x1, x1 y1, y2 = y2 - q * y1, y1 THE RECONSTRUCTION PROBLEM Deﬁnition Sei ein Graph G mit Eckenmenge V = V (G) = {v1,,vn} gegeben.Mit G − vi sei der Untergraph bezeichnet, der durch das L¨oschen der Ecke vi und aller inzidenten Kanten entsteht (also G − vi = G).Die Graphen G − vi heißen vertex-deleted subgraph### von G.

POINT-AT-INFINITY.ORGTRANSLATE THIS PAGE point-at-infinity.org SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page Please do not paste your bitcoin secret keys or other sensible data into the ssss demo page. Observe that there are several independent security issues with that: POINT-AT-INFINITY.ORG Created Date: 2/13/2003 11:47:00 AM PIC00 - 0POINTER.NET pic00 (1/30) Index > > POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > > POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > >### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. POINT-AT-INFINITY.ORG http://point-at-infinity.org/ecc/nisttv Test vectors for the NIST elliptic curves P192, P224, P256, P384, P521, B163, B233, B283, B409, B571, K163, K233, K283, K409### SHANKS & TONELLI

Shanks & Tonelli Pseudocode for the method of Shanks and Tonelli for the computation of square roots in respect to prime moduli is### presented below.

PRIME CURVE JACOBIAN COORDINATES Prime Curve Jacobian Coordinates Introduction 'Jacobian Coordinates' are used to represent elliptic curve points on prime curves y^2 = x^3### + ax + b.

### MODULAR INVERSION

Modular Inversion Python code for two algorithms to compute the inverse of an arbitrary element of a prime field is presented below. The first is an implementation of algorithm 2.107 in the Handbook of Applied Cryptography. def inverse_mod(x, p): assert 0 x p a, b, x2, x1, y2, y1 = p, x, 1, 0, 0, 1 while b > 0: , a = divmod(a, b), b x1, x2 = x2 - q * x1, x1 y1, y2 = y2 - q * y1, y1 THE RECONSTRUCTION PROBLEM Deﬁnition Sei ein Graph G mit Eckenmenge V = V (G) = {v1,,vn} gegeben.Mit G − vi sei der Untergraph bezeichnet, der durch das L¨oschen der Ecke vi und aller inzidenten Kanten entsteht (also G − vi = G).Die Graphen G − vi heißen vertex-deleted subgraph### von G.

POINT-AT-INFINITY.ORGTRANSLATE THIS PAGE point-at-infinity.org SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page Please do not paste your bitcoin secret keys or other sensible data into the ssss demo page. Observe that there are several independent security issues with that: POINT-AT-INFINITY.ORG Created Date: 2/13/2003 11:47:00 AM PIC00 - 0POINTER.NET pic00 (1/30) Index > > POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!) JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > > AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!) JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS: SHAMIR'S SECRET SHARING SCHEME S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on PIC08 - POINT-AT-INFINITY.ORG pic08 (9/30) Index > > AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code!### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. POINT-AT-INFINITY.ORG http://point-at-infinity.org/ecc/nisttv Test vectors for the NIST elliptic curves P192, P224, P256, P384, P521, B163, B233, B283, B409, B571, K163, K233, K283, K409### SHANKS & TONELLI

Shanks & Tonelli Pseudocode for the method of Shanks and Tonelli for the computation of square roots in respect to prime moduli is### presented below.

PRIME CURVE JACOBIAN COORDINATES Prime Curve Jacobian Coordinates Introduction 'Jacobian Coordinates' are used to represent elliptic curve points on prime curves y^2 = x^3### + ax + b.

### MODULAR INVERSION

Modular Inversion Python code for two algorithms to compute the inverse of an arbitrary element of a prime field is presented below. The first is an implementation of algorithm 2.107 in the Handbook of Applied Cryptography. def inverse_mod(x, p): assert 0 x p a, b, x2, x1, y2, y1 = p, x, 1, 0, 0, 1 while b > 0: , a = divmod(a, b), b x1, x2 = x2 - q * x1, x1 y1, y2 = y2 - q * y1, y1 THE RECONSTRUCTION PROBLEM Deﬁnition Sei ein Graph G mit Eckenmenge V = V (G) = {v1,,vn} gegeben.Mit G − vi sei der Untergraph bezeichnet, der durch das L¨oschen der Ecke vi und aller inzidenten Kanten entsteht (also G − vi = G).Die Graphen G − vi heißen vertex-deleted subgraph### von G.

POINT-AT-INFINITY.ORGTRANSLATE THIS PAGE point-at-infinity.org SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page Please do not paste your bitcoin secret keys or other sensible data into the ssss demo page. Observe that there are several independent security issues with that: POINT-AT-INFINITY.ORG Created Date: 2/13/2003 11:47:00 AM PIC00 - 0POINTER.NET pic00 (1/30) Index > > POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEMESECRET SHARING SCHEMESHAMIR SECRET SHARING ALGORITHMSHAMIR SECRET SHARING CALCULATORSECRET SHARING SITESECRET SCHEMETHRESHOLD SECRET SHARING S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into### INDEX OF PICTURES

Index of Pictures : created with igaligal POINT-AT-INFINITY.ORG Some of my projects. ssss - Shamir's Secret Sharing Scheme; seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption; avraes - The AES block cipher on AVR controllers; jsaes & jssha256 - The AES block cipher and the SHA256 message digest in JavaScript; A collection of text material concerning elliptic curve cryptography; Personal. Some unsorted academic papers (in German language!)### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. SECCURE ELLIPTIC CURVE CRYPTO UTILITY FOR RELIABLE ENCRYPTION S ECCURE E lliptic C urve C rypto U tility for R eliable E ncryption . The seccure toolset implements a selection of asymmetric algorithms based on elliptic curve cryptography (ECC). In particular it offers public key encryption / decryption, signature generation / verification and rudimentary key establishment.. ECC schemes offer a much better key size to security ratio than classical systems SSSS - POINT-AT-INFINITY.ORGSEE MORE ON POINT-AT-INFINITY.ORG JSSHA256: SHA256 IN JAVASCRIPT jssha256: SHA256 & HMAC-SHA256 in JavaScript jssha256 is a compact JavaScript implementation of the SHA256 secure hash function. HMAC calculation is also available.. Example. The following code example computes the SHA256 hash value of the string 'abc'. SHA256_init(); SHA256_write("abc"); digest = SHA256_finalize(); digest_hex = array_to_hex_string(digest); JSAES: AES IN JAVASCRIPT jsaes: AES in JavaScript jsaes is a compact JavaScript implementation of the AES block cipher.Key lengths of 128, 192 and 256 bits are supported. Example. The following code example enciphers the plaintext block '00 11 22 ..EE FF' with the 256 bit key '00 01 02 .. 1E 1F'.. AES_Init(); var block = new Array(16); for(var i = 0; i < 16; i++) block = 0x11 * i; var key = new Array(32); for(var i AES ON AVR - POINT-AT-INFINITY.ORG The decryption performs in any case slower than the encryption (what is typical for AES). For further information read the comments in the assembler source code! SSSS: SHAMIR'S SECRET SHARING SCHEMESECRET SHARING SCHEMESHAMIR SECRET SHARING ALGORITHMSHAMIR SECRET SHARING CALCULATORSECRET SHARING SITESECRET SCHEMETHRESHOLD SECRET SHARING S hamir's S ecret S haring S cheme What is "Secret Sharing"? Citing from the Wikipedia article about Secret Sharing: . In cryptography, a secret sharing scheme is a method for distributing a secret amongst a group of participants, each of which is allocated a share of the secret. The secret can only be reconstructed when the shares are combined together; individual shares are of no use on SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page This is the demo page of ssss, an implementation of Shamir's Secret Sharing Scheme.On this page you can try out the splitting of a secret into### INDEX OF PICTURES

Index of Pictures : created with igaligal### INTRODUCTION

Introduction. Elliptic curve cryptography is a modern approach to asymmetric cryptography.It has to be considered a strong competitor to the RSA and DL-based (DSA, Diffie-Hellman) public key encryption and signature schemes. An elliptic curves by itself is a special set of points in the 2D plane. POINT-AT-INFINITY.ORG http://point-at-infinity.org/ecc/nisttv Test vectors for the NIST elliptic curves P192, P224, P256, P384, P521, B163, B233, B283, B409, B571, K163, K233, K283, K409 SECCURE - POINT-AT-INFINITY.ORG seccure SECCURE Elliptic Curve Crypto Utility for Reliable Encryption Synopsis seccure-key seccure-encrypt [-c curve PRIME CURVE JACOBIAN COORDINATES Prime Curve Jacobian Coordinates Introduction 'Jacobian Coordinates' are used to represent elliptic curve points on prime curves y^2 = x^3### + ax + b.

### MODULAR INVERSION

Modular Inversion Python code for two algorithms to compute the inverse of an arbitrary element of a prime field is presented below. The first is an implementation of algorithm 2.107 in the Handbook of Applied Cryptography. def inverse_mod(x, p): assert 0 x p a, b, x2, x1, y2, y1 = p, x, 1, 0, 0, 1 while b > 0: , a = divmod(a, b), b x1, x2 = x2 - q * x1, x1 y1, y2 = y2 - q * y1, y1### 0POINTER.NET

v0.5: (January 2006) - introduction of memory locking and tty echo suppression - a buffer overflow was fixed. It seems to be exploitable. v0.4: (October 2005) - the security level now can be chosen freely in the range 8..1024 bits in steps of 8. SSSS: SHAMIR'S SECRET SHARING SCHEME The ssss demo page Please do not paste your bitcoin secret keys or other sensible data into the ssss demo page. Observe that there are several independent security issues with that: POINT-AT-INFINITY.ORGTRANSLATE THIS PAGE point-at-infinity.org POINT-AT-INFINITY.ORG Created Date: 2/13/2003 11:47:00 AM PIC00 - 0POINTER.NET pic00 (1/30) Index > >### SOME OF MY PROJECTS

* ssss - Shamir's Secret### Sharing Scheme

* seccure - SECCURE Elliptic Curve Crypto Utility for Reliable Encryption * avraes - The AES block cipher on AVR controllers### * jsaes & jssha256

- The AES block cipher and the SHA256 message digest in JavaScript * A collection of text material concerning elliptic curve### cryptography

### PERSONAL

* Some unsorted academic papers (in German language!) I wrote during my studies of mathematics and information science. * Me, eating noodles * Me, kayak-diving from the springboard: photo gallery### and avi video

### (11 MB)

------------------------- The text of this page is licensed under the GNU General Public License . Copyright 2005,2006 by B. Poettering (projects AT point-at-infinity.org). Last modified: Sat Nov 11 16:38:57 CET 2006# Details

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