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What you will learn in Elliptic Curve Cryptography for Developers

by Michael Rosing

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Chapter by chapter

Concepts explained in this book

• group orderElliptic curve cardinality is the number of points on an elliptic curve over a specified finite field, including the point at infinity.
• field extensionA field extension is a field that contains a given base field as a subfield and adds elements needed for further algebraic operations.
• irreducible polynomialAn irreducible polynomial is a nonconstant polynomial that cannot be factored into lower-degree nonconstant polynomials over the specified coefficient field.
• elliptic curve over finite fieldAn elliptic curve over a finite field extension is an elliptic curve whose coefficients and points are defined over an extension field of a finite field.
• j-invariantAn invariant of an elliptic curve that classifies its isomorphism class over an algebraic closure up to standard equivalence.
• point orderThe base point order is the smallest positive scalar whose multiple of the designated base point is the point at infinity.
• quadratic residueA polynomial quadratic residue is a polynomial congruent to the square of another polynomial modulo a specified polynomial modulus.
• quadratic arithmetic programA quadratic arithmetic program is a polynomial constraint representation of an arithmetic computation in which the computation's validity is expressed through quadratic polynomial relationships.
• root of unityA root of unity is an element of a field whose positive integer power equals one.
• embedding degreeThe embedding degree is the smallest positive integer k for which a specified subgroup order divides the multiplicative-group order of the extension field of degree k.
• zero-knowledge proofA zero-knowledge proof is a cryptographic protocol that lets a prover demonstrate that a statement is true or that secret information is known without revealing the secret information itself.
• cross termA cross term is a product involving distinct Lagrange interpolant functions that arises when sums of those functions are multiplied.
• multi-signatureA multi-signature is a cryptographic signature scheme in which multiple signers jointly authorize a message or produce a signature representing their participation.
• common reference stringA common reference string is publicly available setup data shared by the prover and verifier in a cryptographic proof system.
• accountable subgroup multi-signatureA digital signature scheme in which a selected subset of registered participants collectively signs a message and the resulting signature identifies that subset for verification.
• Hilbert class polynomialA Hilbert class polynomial is a polynomial whose roots are j-invariants associated with elliptic curves having complex multiplication by a specified imaginary quadratic order.
• elliptic curve point multiplicationElliptic-curve point multiplication multiplies a point by an integer scalar through repeated point addition and doubling.
• base pointA designated elliptic-curve point whose scalar multiples generate the relevant cyclic subgroup.
• toxic wasteToxic waste is secret randomness and intermediate setup data whose disclosure can compromise the security assumptions of a cryptographic reference string.
• elliptic curveElliptic curve mathematics is the study of algebraic curves defined by nonsingular cubic equations and the mathematical structures on their points.
• elliptic curve point additionElliptic-curve point addition combines two curve points according to the curve's group law to produce another curve point.
• membership keyA membership key is a node-specific cryptographic key computed from the node's row of membership-matrix components and used in that node's signature.
• point at infinityThe point at infinity is the distinguished ideal point included in an elliptic curve's group structure and represented here by the coordinate pair (0, 0).
• square rootA square root is an element whose square equals a specified element in a field or other algebraic system.
• twist curveA twist curve is an elliptic curve related to a given elliptic curve by a field transformation while retaining the same j-invariant and related structural properties.
• public keyA public key is a cryptographic value distributed openly and paired with a private key for encryption, signature verification, or related public-key operations.
• quadratic nonresidueA quadratic nonresidue modulo n is a number that is not congruent to the square of any integer modulo n.
• polynomial functionA polynomial is an algebraic expression formed from coefficients, variables, and nonnegative integer powers combined by addition and multiplication.
• finite fieldA finite field is a field containing a finite number of elements, with addition, subtraction, multiplication, and division by nonzero elements defined within the set.
• zk-SNARKA zk-SNARK is a zero-knowledge SNARK that proves knowledge of a witness without revealing the witness.
• Weil pairingThe Weil pairing is a bilinear, alternating map from pairs of elliptic-curve torsion points to roots of unity in a suitable field extension.
• field primeA field prime is the prime modulus that defines the elements and arithmetic of a prime field used for elliptic-curve computations.
• polynomial divisionPolynomial division expresses a dividend as a divisor times a quotient plus a remainder of lower degree than the divisor.
• nondegeneracyThe property that a pairing does not map every admissible pair of inputs to the identity element.
• MQV key exchange algorithmA full authenticated elliptic-curve MQV key-agreement algorithm that combines static and ephemeral key pairs.
• discriminantAn arithmetic value identifying a quadratic order or related algebraic object that determines a Hilbert class polynomial.
• private keyThe secret key in an asymmetric cryptographic key pair that is controlled by its owner and used for signing or decryption operations.
• polynomial resultantThe resultant of two polynomials is an algebraic scalar that vanishes exactly when the polynomials share a root in an appropriate extension field.
• elliptic curve cofactorThe integer ratio of an elliptic-curve group's order to the order of its selected cryptographic subgroup.
• Tate pairingA Tate pairing is a bilinear elliptic-curve pairing that maps suitable point groups into a multiplicative group of a finite-field extension.
• polynomial pseudo-divisionPolynomial pseudo-division is a division procedure that avoids coefficient inverses by scaling intermediate results by powers of the divisor's leading coefficient.
• modulusThe modulus is the positive integer that defines the equivalence classes and wraparound behavior in modular arithmetic.
• rhoThe rho parameter is the ratio of the logarithmic size of the finite-field modulus to the logarithmic size of the relevant subgroup order in pairing-friendly curve construction.
• signature verificationThe cryptographic procedure that checks whether a signature satisfies the verification equation for specified data and public keys.
• lagrange polynomialA Lagrange polynomial is a polynomial basis function that equals one at its associated interpolation point and zero at the other interpolation points.
• polynomial degreeThe degree of a polynomial is the greatest exponent of the variable having a nonzero coefficient.
• cyclic groupA group generated by a single element, so every group element is an integer multiple or power of that generator.
• ephemeral keyA cryptographic key generated for temporary use in a session or protocol execution rather than retained as a long-term identity key.
• proof of knowledgeA proof of knowledge is a cryptographic proof that demonstrates a prover possesses a witness satisfying a specified statement.
• Fermat’s Little TheoremFermat's little theorem states that a number raised to the appropriate power modulo a prime satisfies a fixed congruence, and its field-extension analogue gives the corresponding finite-field exponent relation.
• modular square rootA square root modulo a prime is an element whose square is congruent to a given element modulo a prime modulus.
• elliptic-curve discrete logarithm problemThe elliptic curve discrete logarithm problem is the problem of recovering a scalar from a base point and its scalar multiple on an elliptic curve.
• Euclid's algorithmThe Euclidean algorithm computes the greatest common divisor of two elements by repeatedly replacing a pair with the divisor and remainder.
• Miller's algorithmAn iterative algorithm for computing an elliptic-curve pairing by evaluating the required rational functions through doubling and addition steps.
• prime fieldA prime field is a finite field with a prime number of elements, commonly represented by the integers modulo a prime.
• greatest common divisorThe greatest common divisor of polynomials is a highest-degree polynomial that divides each given polynomial, defined up to multiplication by a nonzero constant.
• trace of FrobeniusThe integer trace parameter that determines the point count of an elliptic curve over a finite field through the relation #E = p + 1 − t.
• PARI/GPlibpari is the programmable library underlying PARI/GP, providing its computational number-theory functionality for use by other programs.
• Schnorr signatureA public-key digital-signature scheme based on a discrete-logarithm proof of knowledge using a commitment, hash-derived challenge, and response.
• digital signatureA digital signature is a cryptographic value produced with a private key that authenticates data and can be checked with the corresponding public key.
• witnessA witness variable is a secret wire value that supplies private information satisfying a proof statement.
• prime numberA prime number is a positive integer greater than one that has no positive divisors other than one and itself.
• polynomial contentThe content of a polynomial is the common factor of its coefficients, typically normalized according to the coefficient domain.
• Tonelli-Shanks algorithmThe Tonelli-Shanks algorithm computes modular square roots, especially modulo an odd prime.
• monic polynomialA monic polynomial is a nonzero polynomial whose leading coefficient is 1.
• polynomial multiplicationPolynomial multiplication multiplies polynomial expressions and combines terms having equal powers.
• G1G1 is the designated elliptic-curve group serving as the first input group in a pairing whose inputs are written G1 × G2.
• G2G2 is the group of elliptic-curve points on the field-extension curve used as one input group for a pairing.
• proof statementA proof statement is the public claim and associated public inputs whose validity a cryptographic proof establishes.
• multiplication tableA lookup table is a precomputed collection of values arranged for retrieval according to an associated key or index.
• ECDSAThe Elliptic Curve Digital Signature Algorithm, a public-key signature scheme that generates and verifies signatures using elliptic-curve group operations and message hashes.
• elliptic-curve pointAn elliptic curve point is a solution point of an elliptic curve equation over a specified field.
• Ben-Or algorithmBen-Or's algorithm is a randomized or coefficient-search procedure for testing whether a polynomial over a finite field is irreducible.
• complex multiplicationComplex multiplication is a method for constructing elliptic curves with specified endomorphism properties using complex-analytic or algebraic number theory.
• wireA wire is a computational-circuit connection that carries an input or computed value between arithmetic gates.
• GNU Multiple Precision Arithmetic LibraryGNU Multiple Precision Arithmetic Library (GMP) is a software library that provides arbitrary-precision arithmetic for integers, rational numbers, and floating-point values.
• wire variableA wire value is the scalar assigned to a variable or wire in an arithmetic circuit or quadratic arithmetic program.
• key exchangeA cryptographic process in which parties establish a common secret over a network without transmitting that secret directly.
• random pointA random point is an elliptic-curve point selected according to a randomization procedure rather than a deterministic fixed choice.
• inverseAn inverse is a polynomial that multiplies with another polynomial to produce the multiplicative identity modulo a specified irreducible polynomial.
• ellinitellinit is a PARI/GP function that initializes an elliptic-curve data structure from curve coefficients and a defining field or modulus.
• square and multiplySquare-and-multiply is an exponentiation algorithm that repeatedly squares and conditionally multiplies according to the binary representation of the exponent.
• extension degreeThe degree of a finite field extension is the exponent relating the extension size p^k to the size p of its base field.
• bilinearityBilinearity is the property that a map is linear in each of its two arguments, converting addition in either argument into multiplication or addition in the output according to the map's codomain operation.
• Diffie–Hellman key exchangeA public-key key-agreement protocol that lets parties establish a shared secret over an insecure network using private values and exchanged public values.
• membership matrixA matrix whose entries encode the key-membership contributions used to derive each subgroup member's membership key.
• public key aggregationThe operation of combining the public keys of selected signers into a single group element by group addition.
• power methodA modular square-root technique that obtains the root by exponentiating when the prime modulus is congruent to 3 modulo 4.
• elliptic-curve cryptographyElliptic curve cryptography is public-key cryptography based on algebraic operations over elliptic curves.
• elliptic curve data embeddingElliptic curve data embedding is a technique for mapping arbitrary data to a point on an elliptic curve by adjusting a candidate x-coordinate until the curve equation has a square-root solution.
• associate value functionAn MQV protocol function that derives an auxiliary value from a public key, including a truncated x-coordinate in the NIST variant.
• identity elementAn identity element is an element that leaves another element unchanged under a specified operation, such as zero for addition or one for multiplication.
• pairing sweepA pairing sweep is a systematic enumeration of parameter values for pairing-friendly curve formulas followed by tests for acceptable prime and size conditions.
• h0H0 is a BLS hash-to-curve function that maps a message to a point in the G1 subgroup.
• h1H1 is a BLS hash function that maps concatenated public keys to an integer modulo the large prime subgroup order.
• resultant algorithmAn algorithm that computes the resultant of two polynomials through content removal, pseudo-division, and coefficient updates.
• security levelA security level is an estimate of the computational work, commonly expressed in bits, required to defeat a cryptographic system.
• subgroup sizeSubgroup size is the size or bit length of a subgroup, often a large prime-order subgroup of an elliptic-curve group.
• ordinary elliptic curveAn elliptic curve in the ordinary class, as distinguished from supersingular elliptic curves by its arithmetic properties over finite fields.
• statement variableA statement variable is a public wire value that specifies the publicly known part of a proof statement.
• elliptic curve groupAn elliptic curve group is a group formed by the points of an elliptic curve under the curve's algebraically defined group operation.
• proverA prover is the party in a proof protocol that claims a statement or knowledge of secret information and supplies evidence for that claim.
• verifierA verifier is the party in a proof protocol that checks whether the prover's claim and evidence satisfy the required conditions.
• double-and-add algorithmThe double-and-add algorithm computes scalar multiplication using repeated point doubling and conditional point addition while processing the scalar's binary representation.
• complex planeThe complex plane is the two-dimensional plane in which complex numbers are represented by their real and imaginary components.
• extended Euclidean algorithmThe extended Euclidean algorithm computes a greatest common divisor together with coefficients expressing it as a linear combination of the input elements.
• rootA polynomial root is a value in the relevant algebraic structure that satisfies a polynomial equation by making it equal to zero.
• mpz_invertmpz_invert is a GMP function that computes a modular multiplicative inverse when one exists.
• mpz_tmpz_t is GMP's type for representing arbitrary-precision integers.
• mod_addmod_add is a book-defined routine that computes the sum of two values modulo a modulus.
• Legendre symbolThe Legendre symbol is a value in {−1, 0, 1} indicating whether an integer is a quadratic residue modulo an odd prime, whether it is divisible by that prime, or whether it is a quadratic nonresidue.
• polynomial normalizationPolynomial normalization scales every coefficient by the inverse of the leading coefficient to make a polynomial monic.
• subset signatureA subset signature is a multi-signature produced by a specified subset of the participants in an accountable subgroup multi-signature scheme.

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