Small memory footprint fast elliptic encryption

• US 7,650,507 B2
• Filed: 01/16/2009
• Issued: 01/19/2010
• Est. Priority Date: 01/07/2005
• Status: Active Grant

• Current Assignee: Apple Inc.
• Original Assignee: Apple Inc.
• Inventors: Crandall, Richard E., Mitchell, Douglas P.
• Primary Examiner(s): Pich; Ponnoreay
• Application Number:
  US12/355,707
• Publication Number:
  US 20090138721A1
• Time in Patent Office: 368 Days
• Field of Search: 713/176, 713/180
• US Class Current: 713/176
• CPC Class Codes: H04L 2209/80 Wireless network architectures or network communication protocols for wireless network security H04W12/00
• CPC Class Codes: H04L 9/3073 involving pairings, e.g. identity based encryption [IBE], bilinear mappings or bilinear pairings, e.g. Weil or Tate pairing
• CPC Class Codes: H04L 9/3247 involving digital signatures
• CPC Class Codes: H04L 9/3271 using challenge-response

Abstract

A method of generating a digital signature includes generating a first random number from a finite field of numbers, and generating field elements defining a first point on an elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the first random number and an initial public point on the elliptic curve. The method continues by generating a product from a field element, a private key, and a second random number received from a challenger seeking verification of a digital signature, and generating a signature component by summing the product and the first random number. The signature component is reduced using one or more modular reduction operations, using a modulus equal to an order of the elliptic curve, and then the reduced signature component and the field elements are sent to the challenger as a digital signature for verification by the challenger.

Claims

1. A digital signature verification system, comprising:

a processor;

a computer readable medium coupled to the processor and having stored thereon instructions, which, when executed by the processor, cause the processor to perform the operations of;

generating a first random number from a finite field of numbers;

sending the first random number to a signing device;

receiving a digital signature packet from the signing device including a first point on an elliptic curve defined over the finite field of numbers and a signature component, wherein the signature component is a function of the first random number and a second random number generated by the signing device;

generating a second point on the elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the signature component and an initial public point on the elliptic curve;

generating a product from the first random number and a field element of the first point;

reducing the product using one or more modular operations, wherein the one or more modular operations are based on a modulus equal to an order of the elliptic curve;

generating a third point on the elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the reduced product and a public point on the elliptic curve representing a public key of the signing device and distinct from the initial public point; and

evaluating an elliptic identity using the first, second and third points to produce a result indicative of either a positive or negative verification of the digital signature packet.

2. The system of claim 1, wherein one or more of the generating steps uses unsigned field arithmetic.

3. The system of claim 2, wherein the unsigned field arithmetic includes modular operations that are performed without explicit divisions.

4. The system of claim 2, wherein subtract operations on a field element of a respective point on the elliptic curve are performed by negating the field element of the respective point using the identity (−

y)mod p w^s−

(y+k)mod p, where y is the field element to be negated, p is a field prime, w is an n-bit segment of memory, k is a positive integer, and s is an exponent used to set a security level.

5. The system of claim 2, wherein the unsigned field arithmetic does not use field inversion operations.

6. The system of claim 2, wherein the field arithmetic does not use dynamically allocated memory.

7. The system of claim 2, wherein a maximum number of bytes of memory allocated for temporary storage of variables used by field arithmetic operations used to verify the digital signature packet does not exceed 226 bytes.

8. The system of claim 2, wherein at least one temporary variable used by the unsigned field arithmetic is shared by at least two field arithmetic operations.

9. The system of claim 2, wherein the digital signature packet received from the signing device is a packet not exceeding 3s+3 memory segments w, wherein s represents the number of n-bit memory segments w used to store unsigned integer values for the signature component and field elements of the first point.

10. The system of claim 2, wherein the computer-readable medium includes a data structure for storing nine parameters used to verify the digital signature packet, wherein eight parameters are stored in no more than s+1 n-bit memory segments w and one parameter is stored in no more than 2s n-bit segments w, where s is a positive integer.

11. The system of claim 10, wherein s is equal to 10 and n is equal to 16.

12. The system of claim 1, wherein the elliptic curve has a curve order o=w^s−

j, wherein w is an n-bit segment of computer memory, and s and j are positive integers.

13. The system of claim 12, wherein the first random number is an integer constrained to reside in the interval [2, o−

1].

14. The system of claim 12, wherein the initial public point on the elliptic curve has a point order that divides the curve order o and contains the largest prime factor in the curve order o.

15. The system of claim 12, wherein j is greater than 0 and less than w^1+s/2.

16. The system of claim 1, wherein field elements of the first, second, and third points are Montgomery coordinate pairs.

17. The system of claim 1, wherein the elliptic curve is defined in an x-y coordinate plane by the equation y²=x³+cx²+x.

18. The system of claim 17, wherein c is equal to 4.

19. The system of claim 1, wherein field elements of the first, second, and third points reside in an interval [0, p−

1], wherein p is a field prime.

20. The system of claim 1, wherein the second random number has a Hamming weight less than or equal to 48.

21. A computer-implemented method of verifying a digital signature, comprising:

at an electronic device comprising one or more processors and a communications interface coupled to a signing device;

generating a first random number from a finite field of numbers;

sending the first random number to a signing device;

receiving a digital signature packet from a signing device including a first point on an elliptic curve defined over the finite field of numbers and a signature component, wherein the signature component is a function of the first random number and a second random number generated by the signing device;

generating a second point on the elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the signature component and an initial public point on the elliptic curve;

generating a product from the first random number and a field element of the first point;

reducing the product using one or more modular operations, wherein the one or more modular operations are based on a modulus equal to an order of the elliptic curve;

generating a third point on the elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the reduced product and a public point on the elliptic curve representing a public key of the signing device and distinct from the initial public point; and

evaluating an elliptic identity using the first, second and third points to produce a result indicative of either a positive or negative verification of the digital signature packet.

22. A computer-readable medium having stored thereon instructions, which, when executed by a processor in a system for generating a digital signature, cause the processor to perform the operations of:

generating a first random number from a finite field of numbers;

sending the first random number to a signing device;

receiving a digital signature packet from a signing device including a first point on an elliptic curve defined over the finite field of numbers and a signature component, wherein the signature component is a function of the first random number and a second random number generated by the signing device;

generating a second point on the elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the signature component and an initial public point on the elliptic curve;

generating a product from the first random number and a field element of the first point;

reducing the product using one or more modular operations, wherein the one or more modular operations are based on a modulus equal to an order of the elliptic curve;

generating a third point on the elliptic curve defined over the finite field of numbers by performing elliptic curve arithmetic on the reduced product and a public point on the elliptic curve representing a public key of the signing device and distinct from the initial public point; and

evaluating an elliptic identity using the first, second and third points to produce a result indicative of either a positive or negative verification of the digital signature packet.